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interruptpari.pspari.log *** Error in the PARI system. End of program. xsegmentation fault: bug in PARI or calling programfloating point exception: bug in PARI or calling programunknown signalstack too largemissing cell in err_catch_stack. Resetting all trapsmallocing NULL object in newblocnew bloc, size %6ld (no %ld): %08lx killing bloc (no %ld): %08lx reordervariable out of range in reorderduplicated indeterminates in reorder *** %s: ... *** For full compatibility with GP 1.39, type "default(compatible,3)" (you can also set "compatible = 3" in your GPRC file)%s file *** %s %s is not yet implemented. in %s. %s, please reportadditionmultiplicationdivision,gcd-->assignment %s %s %s %s.. in %s; new prec = %ld %s: %s current stack size: %ld (%.3f Mbytes) [hint] you can increase GP stack with allocatemem() bad object %Zbad component %ld in object %Zlbot>ltop in gerepiledoubling stack size; new stack = %ld (%.3f Mbytes)required stack memory too smallmallocing NULL objectnot an available timer (%ld)no timers left! 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Empty set if x is omittedStr({x=""},{flag=0}): transforms any GEN x into a string. Empty string if x is omitted. If flag is set, perform tilde expansion on stringVec({x=[]}): transforms the object x into a vector. Used mainly if x is a polynomial or a power series. Empty vector if x is omittedVecsmall({x=[]}): transforms the object x into a VECSMALL. Empty vector if x is omittedabs(x): absolute value (or modulus) of xacos(x): inverse cosine of xacosh(x): inverse hyperbolic cosine of xaddprimes({x=[]}): add primes in the vector x (with at most 100 components) to the prime table. x may also be a single integer. The "primes" may in fact be composite, obtained for example by the function factor(x,0), and in that case the message "impossible inverse modulo" will give you some factors. List the current extra primes if x is omitted. If some primes are added which divide non trivially the existing table, suitable updating is doneagm(x,y): arithmetic-geometric mean of x and yalgdep(x,n,{flag=0}): algebraic relations up to degree n of x. flag is optional, and can be 0: default, uses the algorithm of Hastad et al, or non-zero, and in that case is a number of decimal digits which should be between 0.5 and 1.0 times the number of decimal digits of accuracy of x, and uses a standard LLLalias("new","old"): new is now an alias for oldarg(x): argument of x,such that -pi0. The answer is guaranteed (i.e x norm iff b=1) under GRH, if S contains all primes less than 12.log(disc(Bnf))^2, where Bnf is the Galois closure of bnfbnfisprincipal(bnf,x,{flag=1}): bnf being output by bnfinit (with flag<=2), gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vector. flag is optional, whose meaning is: 0: output only v; 1: default; 2: output only v, precision being doubled until the result is obtained; 3: as 2 but output generatorsbnfissunit(bnf,sfu,x): bnf being output by bnfinit (with flag<=2), sfu by bnfsunit, gives the column vector of exponents of x on the fundamental S-units and the roots of unity if x is a unit, the empty vector otherwisebnfisunit(bnf,x): bnf being output by bnfinit (with flag<=2), gives the column vector of exponents of x on the fundamental units and the roots of unity if x is a unit, the empty vector otherwisebnfmake(sbnf): transforms small sbnf as output by bnfinit with flag=3 into a true big bnfbnfnarrow(bnf): given a big number field as output by bnfinit, gives as a 3-component vector the structure of the narrow class groupbnfreg(P,{tech=[]}): compute the regulator of the number field defined by the polynomial P. If P is a non-zero integer, it is interpreted as a quadratic discriminant. See manual for details about techbnfsignunit(bnf): matrix of signs of the real embeddings of the system of fundamental units found by bnfinitbnfsunit(bnf,S): compute the fundamental S-units of the number field bnf output by bnfinit, S being a list of prime ideals. res[1] contains the S-units, res[5] the S-classgroup. See manual for detailsbnfunit(bnf): compute the fundamental units of the number field bnf output by bnfinit when they have not yet been computed (i.e. with flag=2)bnrL1(bnr, subgroup, {flag=0}): bnr being output by bnrinit(,,1) and subgroup being a square matrix defining a congruence subgroup of bnr (or 0 for the trivial subgroup), for each character of bnr trivial on this subgroup, compute L(1, chi) (or equivalently the first non-zero term c(chi) of the expansion at s = 0). The binary digits of flag mean 1: if 0 then compute the term c(chi) and return [r(chi), c(chi)] where r(chi) is the order of L(s, chi) at s = 0, or if 1 then compute the value at s = 1 (and in this case, only for non-trivial characters), 2: if 0 then compute the value of the primitive L-function associated to chi, if 1 then compute the value of the L-function L_S(s, chi) where S is the set of places dividing the modulus of bnr (and the infinite places), 3: return also the charactersbnrclass(bnf,ideal,{flag=0}): given a big number field as output by bnfinit (only) and an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, finds the ray class group structure corresponding to this module. flag is optional, and can be 0: default, 1: compute data necessary for working in the ray class group, for example with functions such as bnrisprincipal or bnrdisc, without computing the generators of the ray class group, or 2: with the generators. When flag=1 or 2, the fifth component is the ray class group structure obtained when flag=0bnrclassno(bnf,x): ray class number of the module x for the big number field bnf. Faster than bnrclass if only the ray class number is wantedbnrclassnolist(bnf,list): if list is as output by ideallist or similar, gives list of corresponding ray class numbersbnrconductor(a1,{a2},{a3},{flag=0}): conductor of the subfield of the ray class field given by a1,a2,a3 (see bnrdisc). flag is optional and can be 0: default, or nonzero positive: returns [conductor,rayclassgroup,subgroup], or nonzero negative: returns 1 if modulus is the conductor and 0 otherwise (same as bnrisconductor)bnrconductorofchar(bnr,chi): conductor of the character chi on the ray class group bnrbnrdisc(a1,{a2},{a3},{flag=0}): absolute or relative [N,R1,discf] of the field defined by a1,a2,a3. [a1,{a2},{a3}] is of type [bnr], [bnr,subgroup], [bnf, module] or [bnf,module,subgroup], where bnf is as output by bnfclassunit (with flag<=2), bnr by bnrclass (with flag>0), and subgroup is the HNF matrix of a subgroup of the corresponding ray class group (if omitted, the trivial subgroup). flag is optional whose binary digits mean 1: give relative data; 2: return 0 if module is not the conductorbnrdisclist(bnf,bound,{arch},{flag=0}): gives list of discriminants of ray class fields of all conductors up to norm bound, where the ramified Archimedean places are given by arch (unramified at infinity if arch is void), in a long vector format. If (optional) flag is present and non-null, give arch all the possible values. Supports the alternative syntax bnrdisclist(bnf,list), where list is as output by ideallist or ideallistarch (with units)bnrinit(bnf,ideal,{flag=0}): given a big number field as output by bnfinit (only) and an ideal or a 2-component row vector formed by an ideal and a list of R1 zeros or ones representing a module, initializes data linked to the ray class group structure corresponding to this module. flag is optional, and can be 0: default (same as bnrclass with flag = 1), 1: compute also the generators (same as bnrclass with flag = 2). The fifth component is the ray class group structurebnrisconductor(a1,{a2},{a3}): returns 1 if the modulus is the conductor of the subfield of the ray class field given by a1,a2,a3 (see bnrdisc), and 0 otherwise. Slightly faster than bnrconductor if this is the only desired resultbnrisprincipal(bnr,x,{flag=1}): bnr being output by bnrinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vector. If (optional) flag is set to 0, output only vbnrrootnumber(bnr,chi,{flag=0}); returns the so-called Artin Root Number, i.e. the constant W appearing in the functional equation of the Hecke L-function associated to chi. Set flag = 1 if the character is known to be primitivebnrstark(bnr,subgroup,{flag=0}): bnr being as output by bnrinit(,,1), finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup using Stark's units. The ground field and the class field must be totally real. flag is optional and may be set to 0 to obtain a reduced polynomial, 1 to obtain a non reduced polynomial, 2 to obtain an absolute polynomial and 3 to obtain the irreducible polynomial of the Stark unit, 0 being default. If 4 is added to the value of flag, try hard to find the best modulusbreak({n=1}): interrupt execution of current instruction sequence, and exit from the n innermost enclosing loopsceil(x): ceiling of x=smallest integer>=xcenterlift(x,{v}): centered lift of x. Same as lift except for integermodschangevar(x,y): change variables of x according to the vector ycharpoly(A,{v=x},{flag=0}): det(v*Id-A)=characteristic polynomial of the matrix A using the comatrix. flag is optional and may be set to 1 (use Lagrange interpolation) or 2 (use Hessenberg form), 0 being the defaultchinese(x,y): x,y being integers modulo mx and my, finds z such that z is congruent to x mod mx and y mod mycomponent(x,s): the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]. For list objects such as nf, bnf, bnr or ell, it is much easier to use member functions starting with "." concat(x,{y}): concatenation of x and y, which can be scalars, vectors or matrices, or lists (in this last case, both x and y have to be lists). If y is omitted, x has to be a list or row vector and its elements are concatenatedconj(x): the algebraic conjugate of xconjvec(x): conjugate vector of the algebraic number xcontent(x): gcd of all the components of x, when this makes sensecontfrac(x,{b},{lmax}): continued fraction expansion of x (x rational,real or rational function). b and lmax are both optional, where b is the vector of numerators of the continued fraction, and lmax is a bound for the number of terms in the continued fraction expansioncontfracpnqn(x): [p_n,p_{n-1}; q_n,q_{n-1}] corresponding to the continued fraction xcore(n,{flag=0}): unique (positive of negative) squarefree integer d dividing n such that n/d is a square. If (optional) flag is non-null, output the two-component row vector [d,f], where d is the unique squarefree integer dividing n such that n/d=f^2 is a squarecoredisc(n,{flag=0}): discriminant of the quadratic field Q(sqrt(n)). If (optional) flag is non-null, output a two-component row vector [d,f], where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f may be a half integercos(x): cosine of xcosh(x): hyperbolic cosine of xcotan(x): cotangent of xdenominator(x): denominator of x (or lowest common denominator in case of an array)deriv(x,{y}): derivative of x with respect to the main variable of y, or to the main variable of x if y is omitteddilog(x): dilogarithm of xdirdiv(x,y): division of the Dirichlet series x by the Dirichlet series ydireuler(p=a,b,expr,{c}): Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms. Expr should be a polynomial or rational function in p and X, and X is understood to mean p^(-s). If c is present, output only the first c termsdirmul(x,y): multiplication of the Dirichlet series x by the Dirichlet series ydirzetak(nf,b): Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1divisors(x): gives a vector formed by the divisors of x in increasing orderdivrem(x,y): euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remaindereint1(x,{n}): exponential integral E1(x). If n is present, computes the vector of the first n values of the exponential integral E1(n.x) (x > 0)elladd(e,z1,z2): sum of the points z1 and z2 on elliptic curve eellak(e,n): computes the n-th Fourier coefficient of the L-function of the elliptic curve eellan(e,n): computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<2^24 on a 32-bit machine)ellap(e,p,{flag=0}): computes a_p for the elliptic curve e using Shanks-Mestre's method. flag is optional and can be set to 0 (default) or 1 (use Jacobi symbols)ellbil(e,z1,z2): canonical bilinear form for the points z1,z2 on the elliptic curve e. Either z1 or z2 can also be a vector/matrix of pointsellchangecurve(x,y): change data on elliptic curve according to y=[u,r,s,t]ellchangepoint(x,y): change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]elleisnum(om,k,{flag=0}): om=[om1,om2] being a 2-component vector giving a basis of a lattice L and k an even positive integer, computes the numerical value of the Eisenstein series of weight k. When flag is non-zero and k=4 or 6, this gives g2 or g3 with the correct normalizationelleta(om): om=[om1,om2], returns the two-component row vector [eta1,eta2] of quasi-periods associated to [om1,om2]ellglobalred(e): e being an elliptic curve, returns [N,[u,r,s,t],c], where N is the conductor of e, [u,r,s,t] leads to the standard model for e, and c is the product of the local Tamagawa numbers c_pellheight(e,x,{flag=0}): canonical height of point x on elliptic curve E defined by the vector e. flag is optional and should be 0 or 1 (0 by default): 0: use theta-functions, 1: use Tate's methodellheightmatrix(e,x): gives the height matrix for vector of points x on elliptic curve e using theta functionsellinit(x,{flag=0}): x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j,[e1,e2,e3],w1,w2,eta1,eta2,area]. If the curve is defined over a p-adic field, the last six components are replaced by root,u^2,u,q,w,0. If optional flag is 1, omit them altogetherellisoncurve(e,x): true(1) if x is on elliptic curve e, false(0) if notellj(x): elliptic j invariant of xelllocalred(e,p): e being an elliptic curve, returns [f,kod,[u,r,s,t],c], where f is the conductor's exponent, kod is the kodaira type for e at p, [u,r,s,t] is the change of variable needed to make e minimal at p, and c is the local Tamagawa number c_pelllseries(e,s,{A=1}): L-series at s of the elliptic curve e, where A a cut-off point close to 1ellorder(e,p): order of the point p on the elliptic curve e over Q, 0 if non-torsionellordinate(e,x): y-coordinates corresponding to x-ordinate x on elliptic curve eellpointtoz(e,P): lattice point z corresponding to the point P on the elliptic curve eellpow(e,x,n): n times the point x on elliptic curve e (n in Z)ellrootno(e,{p=1}): root number for the L-function of the elliptic curve e. p can be 1 (default), global root number, or a prime p (including 0) for the local root number at pellsigma(om,z,{flag=0}): om=[om1,om2], value of the Weierstrass sigma function of the lattice generated by om at z if flag = 0 (default). If flag = 1, arbitrary determination of the logarithm of sigma. If flag = 2 or 3, same but using the product expansion instead of theta seriesellsub(e,z1,z2): difference of the points z1 and z2 on elliptic curve eelltaniyama(e): modular parametrization of elliptic curve eelltors(e,{flag=0}): torsion subgroup of elliptic curve e: order, structure, generators. If flag = 0, use Doud's algorithm; if flag = 1, use Lutz-Nagellellwp(e,{z=x},{flag=0}): Complex value of Weierstrass P function at z on the lattice generated over Z by e=[om1,om2] (e as given by ellinit is also accepted). Optional flag means 0 (default), compute only P(z), 1 compute [P(z),P'(z)], 2 consider om as an elliptic curve and compute P(z) for that curve (identical to ellztopoint in that case). If z is omitted or is a simple variable, return formal expansion in zellzeta(om,z): om=[om1,om2], value of the Weierstrass zeta function of the lattice generated by om at zellztopoint(e,z): coordinates of point P on the curve e corresponding to the complex number zerfc(x): complementary error functioneta(x,{flag=0}): if flag=0, eta function without the q^(1/24), otherwise eta of the complex number x in the upper half plane intelligently computed using SL(2,Z) transformationseulerphi(x): Euler's totient function of xeval(x): evaluation of x, replacing variables by their valueexp(x): exponential of xfactor(x,{lim}): factorization of x. lim is optional and can be set whenever x is of (possibly recursive) rational type. If lim is set return partial factorization, using primes up to lim (up to primelimit if lim=0)factorback(fa,{nf}): given a factorisation fa, gives the factored object back. If this is a prime ideal factorisation you must supply the corresponding number field as second argumentfactorcantor(x,p): factorization mod p of the polynomial x using Cantor-Zassenhausfactorff(x,p,a): factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]factorial(x): factorial of x (x C-integer), the result being given as a real numberfactorint(x,{flag=0}): factor the integer x. flag is optional, whose binary digits mean 1: avoid MPQS, 2: avoid first-stage ECM (may fall back on it later), 4: avoid Pollard-Brent Rho and Shanks SQUFOF, 8: skip final ECM (huge composites will be declared prime)factormod(x,p,{flag=0}): factorization mod p of the polynomial x using Berlekamp. flag is optional, and can be 0: default or 1: simple factormod, same except that only the degrees of the irreducible factors are givenfactornf(x,t): factorization of the polynomial x over the number field defined by the polynomial tfactorpadic(x,p,r,{flag=0}): p-adic factorization of the polynomial x to precision r. flag is optional and may be set to 0 (use round 4) or 1 (use Buchmann-Lenstra)ffinit(p,n,{v=x}): monic irreducible polynomial of degree n over F_p[v]fibonacci(x): fibonacci number of index x (x C-integer)floor(x): floor of x = largest integer<=xfor(X=a,b,seq): the sequence is evaluated, X going from a up to bfordiv(n,X,seq): the sequence is evaluated, X running over the divisors of nforprime(X=a,b,seq): the sequence is evaluated, X running over the primes between a and bforstep(X=a,b,s,seq): the sequence is evaluated, X going from a to b in steps of s (can be a vector of steps)forsubgroup(H=G,{bound},seq): execute seq for each subgroup H of the abelian group G (in SNF form), whose index is bounded by bound. H is given as a left divisor of G in HNF formforvec(x=v,seq,{flag=0}): v being a vector of two-component vectors of length n, the sequence is evaluated with x[i] going from v[i][1] to v[i][2] for i=n,..,1 if flag is zero or omitted. If flag = 1 (resp. flag = 2), restrict to increasing (resp. strictly increasing) sequencesfrac(x): fractional part of x = x-floor(x)galoisfixedfield(gal,perm,{flag},{v=y}): gal being a galois field as output by galoisinit and perm an element of gal.group or a vector of such elements, return [P,x] such that P is a polynomial defining the fixed field of gal[1] by the subgroup generated by perm, and x is a root of P in gal expressed as a polmod in gal.pol. If flag is 1 return only P. If flag is 2 return [P,x,F] where F is the factorization of gal.pol over the field defined by P, where the variable v stands for a root of Pgaloisinit(pol,{den}): pol being a polynomial or a number field as output by nfinit defining a Galois extension of Q, compute the Galois group and all neccessary informations for computing fixed fields. den is optional and has the same meaning as in nfgaloisconj(,4)(see manual)galoispermtopol(gal,perm): gal being a galois field as output by galoisinit and perm a element of gal.group, return the polynomial defining the corresponding Galois automorphismgaloissubcyclo(n,H,{Z},{v},{fl=0}):Compute a polynomial defining the subfield of Q(zeta_n) fixed by the subgroup H of Z/nZ. H can be given by a generator, a set of generator given by a vector or a SNF matrix. If present Z must be znstar(n), currently it is used only when H is a SNF matrix. If v is given, the polynomial is given in the variable v. If flag is 1, output only the conductor of the abelian extension. If flag is 2 output [pol,f] where pol is the polynomial and f the conductor.gamma(x): gamma function at xgammah(x): gamma of x+1/2 (x integer)gcd(x,y,{flag=0}): greatest common divisor of x and y. flag is optional, and can be 0: default, 1: use the modular gcd algorithm (x and y must be polynomials), 2 use the subresultant algorithm (x and y must be polynomials)getheap(): 2-component vector giving the current number of objects in the heap and the space they occupygetrand(): current value of random number seedgetstack(): current value of stack pointer avmagettime(): time (in milliseconds) since last call to gettimehilbert(x,y,{p}): Hilbert symbol at p of x,y. If x,y are integermods or p-adic, p can be omittedhyperu(a,b,x): U-confluent hypergeometric functionidealadd(nf,x,y): sum of two ideals x and y in the number field defined by nfidealaddtoone(nf,x,{y}): if y is omitted, when the sum of the ideals in the number field K defined by nf and given in the vector x is equal to Z_K, gives a vector of elements of the corresponding ideals who sum to 1. Otherwise, x and y are ideals, and if they sum up to 1, find one element in each of them such that the sum is 1idealappr(nf,x,{flag=0}): x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other p. If (optional) flag is non-null x must be a prime ideal factorization with possibly zero exponentsidealchinese(nf,x,y): x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealcoprime(nf,x,y): gives an element b in nf such that b. x is an integral ideal coprime to the integral ideal yidealdiv(nf,x,y,{flag=0}): quotient x/y of two ideals x and y in HNF in the number field nf. If (optional) flag is non-null, the quotient is supposed to be an integral ideal (slightly faster)idealfactor(nf,x): factorization of the ideal x given in HNF into prime ideals in the number field nfidealhnf(nf,a,{b}): hermite normal form of the ideal a in the number field nf, whatever form a may have. If called as idealhnf(nf,a,b), the ideal is given as aZ_K+bZ_K in the number field K defined by nfidealintersect(nf,x,y): intersection of two ideals x and y in the number field defined by nfidealinv(nf,x,{flag=0}): inverse of the ideal x in the number field nf. If flag is omitted or set to 0, use the different. If flag is 1 do not use itideallist(nf,bound,{flag=4}): vector of vectors L of all idealstar of all ideals of norm<=bound. If (optional) flag is present, its binary digits are toggles meaning 1: give generators; 2: output [L,U], where L is as before, and U is a vector of vector of zinternallogs of the units; 4: give only the ideals and not the idealstarideallistarch(nf,list,{arch=[]},{flag=0}): vector of vectors of all idealstarinit of all modules in list with archimedean arch (void if ommited or arch=[]) added. flag is optional whose binary digits are toggles meaning 1: give generators as well; 2: list format is [L,U], see ideallistideallog(nf,x,bid): if bid is a big ideal, as given by idealstar(nf,I,1) or idealstar(nf,I,2), gives the vector of exponents on the generators bid[2][3] (even if these generators have not been computed)idealmin(nf,ix,{vdir}): minimum of the ideal ix in the direction vdir in the number field nfidealmul(nf,x,y,{flag=0}): product of the two ideals x and y in the number field nf. If (optional) flag is non-nul, reduce the resultidealnorm(nf,x): norm of the ideal x in the number field nfidealpow(nf,x,n,{flag=0}): n-th power of the ideal x in HNF in the number field nf If (optional) flag is non-null, reduce the resultidealprimedec(nf,p): prime ideal decomposition of the prime number p in the number field nf as a vector of 5 component vectors [p,a,e,f,b] representing the prime ideals pZ_K+a. Z_K, e,f as usual, a as vector of components on the integral basis, b Lenstra's constantidealprincipal(nf,x): returns the principal ideal generated by the algebraic number x in the number field nfidealred(nf,x,{vdir=0}): LLL reduction of the ideal x in the number field nf along direction vdir, in HNFidealstar(nf,I,{flag=1}): gives the structure of (Z_K/I)^*. flag is optional, and can be 0: simply gives the structure as a 3-component vector v such that v[1] is the order (i.e. eulerphi(I)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generators. If flag=1 (default), gives idealstarinit, i.e. a 6-component vector [I,v,fa,f2,U,V] where v is as above without the generators, fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*. Finally if flag=2, same as with flag=1 except that the generators are also givenidealtwoelt(nf,x,{a}): two-element representation of an ideal x in the number field nf. If (optional) a is non-zero, first element will be equal to aidealval(nf,x,p): valuation at p given in idealprimedec format of the ideal x in the number field nfideleprincipal(nf,x): returns the principal idele generated by the algebraic number x in the number field nfif(a,seq1,seq2): if a is nonzero, seq1 is evaluated, otherwise seq2. seq1 and seq2 are optional, and if seq2 is omitted, the preceding comma can be omitted alsoimag(x): imaginary part of xincgam(s,x,{y}): incomplete gamma function. y is optional and is the precomputed value of gamma(s)incgamc(s,x): complementary incomplete gamma functionintformal(x,{y}): formal integration of x with respect to the main variable of y, or to the main variable of x if y is omittedintnum(X=a,b,s,{flag=0}): numerical integration of s (smooth in ]a,b[) from a to b with respect to X. flag is optional and mean 0: default. s can be evaluated exactly on [a,b]; 1: general function; 2: a or b can be plus or minus infinity (chosen suitably), but of same sign; 3: s has only limits at a or bisfundamental(x): true(1) if x is a fundamental discriminant (including 1), false(0) if notisprime(x,{flag=0}): if flag is omitted or 0 true(1) if x is a strong pseudoprime for 10 random bases, false(0) if not. If flag is 1 the primality is certified by Pocklington-Lehmer Test. If flag is 2 a primality certificate is output(see manual)ispseudoprime(x): true(1) if x is a strong pseudoprime, false(0) if notissquare(x,{&n}): true(1) if x is a square, false(0) if not. If n is given puts the exact square root there if it was computedissquarefree(x): true(1) if x is squarefree, false(0) if notkronecker(x,y): kronecker symbol (x/y)lcm(x,y): least common multiple of x and y=x*y/gcd(x,y)length(x): number of non code words in x, number of characters for a stringlex(x,y): compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x=0, transforms the rational or integral mxn (m>=n) matrix x into an integral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not divisible by p otherwise. If p=-1, finds a basis of the intersection with Z^n of the lattice spanned by the columns of x. If p=-2, finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xmatsize(x): number of rows and columns of the vector/matrix x as a 2-vectormatsnf(x,{flag=0}): Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vector d. Binary digits of flag mean 1: returns [u,v,d] where d=u*x*v, otherwise only the diagonal d is returned, 2: allow polynomial entries, otherwise assume x is integral, 4: removes all information corresponding to entries equal to 1 in dmatsolve(M,B): gaussian solution of MX=B (M matrix, B column vector)matsolvemod(M,D,B,{flag=0}): one solution of system of congruences MX=B mod D (M matrix, B and D column vectors). If (optional) flag is non-null return all solutionsmatsupplement(x): supplement the columns of the matrix x to an invertible matrixmattranspose(x): x~=transpose of xmax(x,y): maximum of x and ymin(x,y): minimum of x and ymodreverse(x): reverse polymod of the polymod x, if it existsmoebius(x): Moebius function of xnewtonpoly(x,p): Newton polygon of polynomial x with respect to the prime pnext({n=1}): interrupt execution of current instruction sequence, and start another iteration from the n-th innermost enclosing loopsnextprime(x): smallest prime number>=xnfalgtobasis(nf,x): transforms the algebraic number x into a column vector on the integral basis nf.zknfbasis(x,{flag=0},{p}): integral basis of the field Q[a], where a is a root of the polynomial x, using the round 4 algorithm. Second and third args are optional. Binary digits of flag mean 1: assume that no square of a prime>primelimit divides the discriminant of x, 2: use round 2 algorithm instead. If present, p provides the matrix of a partial factorization of the discriminant of x, useful if one wants only an order maximal at certain primes onlynfbasistoalg(nf,x): transforms the column vector x on the integral basis into an algebraic numbernfdetint(nf,x): multiple of the ideal determinant of the pseudo generating set xnfdisc(x,{flag=0},{p}): discriminant of the number field defined by the polynomial x using round 4. Optional args flag and p are as in nfbasisnfeltdiv(nf,a,b): element a/b in nfnfeltdiveuc(nf,a,b): gives algebraic integer q such that a-bq is smallnfeltdivmodpr(nf,a,b,pr): element a/b modulo pr in nf, where pr is in prhall format (see nfmodprinit)nfeltdivrem(nf,a,b): gives [q,r] such that r=a-bq is smallnfeltmod(nf,a,b): gives r such that r=a-bq is small with q algebraic integernfeltmul(nf,a,b): element a. b in nfnfeltmulmodpr(nf,a,b,pr): element a. b modulo pr in nf, where pr is in prhall format (see nfmodprinit)nfeltpow(nf,a,k): element a^k in nfnfeltpowmodpr(nf,a,k,pr): element a^k modulo pr in nf, where pr is in prhall format (see nfmodprinit)nfeltreduce(nf,a,id): gives r such that a-r is in the ideal id and r is smallnfeltreducemodpr(nf,a,pr): element a modulo pr in nf, where pr is in prhall format (see nfmodprinit)nfeltval(nf,a,pr): valuation of element a at the prime pr as output by idealprimedecnffactor(nf,x): factor polynomial x in number field nfnffactormod(nf,pol,pr): factorize polynomial pol modulo prime ideal pr in number field nfnfgaloisapply(nf,aut,x): Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfnfgaloisconj(nf,{flag=0},{den}): list of conjugates of a root of the polynomial x=nf.pol in the same number field. flag is optional (set to 0 by default), meaning 0: use combination of flag 4 and 1, always complete; 1: use nfroots; 2 : use complex numbers, LLL on integral basis (not always complete); 4: use Allombert's algorithm, complete if the field is Galois of degree <= 35 (see manual for detail). nf can be simply a polynomial with flag 0,2 and 4, meaning: 0: use combination of flag 4 and 2, not always complete (but a warning is issued when the list is not proven complete); 2 & 4: same meaning and restrictions. Note that only flag 4 can be applied to fields of large degrees (approx. >= 20)nfhilbert(nf,a,b,{p}): if p is omitted, global Hilbert symbol (a,b) in nf, that is 1 if X^2-aY^2-bZ^2 has a non-trivial solution (X,Y,Z) in nf, -1 otherwise. Otherwise compute the local symbol modulo the prime ideal pnfhnf(nf,x): if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfhnfmod(nf,x,detx): if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_jnfinit(pol,{flag=0}): pol being a nonconstant irreducible polynomial, gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual),r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]. flag is optional and can be set to 0: default; 1: do not compute different; 2: first use polred to find a simpler polynomial; 3: outputs a two-element vector [nf,Mod(a,P)], where nf is as in 2 and Mod(a,P) is a polymod equal to Mod(x,pol) and P=nf.pol; 4: as 2 but use a partial polred; 5: is to 3 what 4 is to 2nfisideal(nf,x): true(1) if x is an ideal in the number field nf, false(0) if notnfisincl(x,y): tests whether the number field x is isomorphic to a subfield of y (where x and y are either polynomials or number fields as output by nfinit). Return 0 if not, and otherwise all the isomorphisms. If y is a number field, a faster algorithm is usednfisisom(x,y): as nfisincl but tests whether x is isomorphic to ynfkermodpr(nf,x,pr): kernel of the matrix x in Z_K/pr, where pr is in prhall format (see nfmodprinit)nfmodprinit(nf,pr): transform the 5 element row vector pr representing a prime ideal into prhall format necessary for all operations mod pr in the number field nf (see manual for details about the format)nfnewprec(nf): transform the number field data nf into new data using the current (usually larger) precisionnfroots(nf,pol): roots of polynomial pol belonging to nf without multiplicitynfrootsof1(nf): number of roots of unity and primitive root of unity in the number field nfnfsnf(nf,x): if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xnfsolvemodpr(nf,a,b,pr): solution of a*x=b in Z_K/pr, where a is a matrix and b a column vector, and where pr is in prhall format (see nfmodprinit)nfsubfields(nf,{d=0}): find all subfields of degree d of number field nf (all subfields if d is null or omitted). Result is a vector of subfields, each being given by [g,h], where g is an absolute equation and h expresses one of the roots of g in terms of the root x of the polynomial defining nfnorm(x): norm of xnorml2(x): square of the L2-norm of the vector xnumdiv(x): number of divisors of xnumerator(x): numerator of xnumtoperm(n,k): permutation number k (mod n!) of n letters (n C-integer)omega(x): number of distinct prime divisors of xpadicappr(x,a): p-adic roots of the polynomial x congruent to a mod ppadicprec(x,p): absolute p-adic precision of object xpermtonum(vect): ordinal (between 1 and n!) of permutation vectpolcoeff(x,s,{v}): coefficient of degree s of x, or the s-th component for vectors or matrices (for which it is simpler to use x[]). With respect to the main variable if v is omitted, with respect to the variable v otherwisepolcompositum(pol1,pol2,{flag=0}): vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2. If (optional) flag is set (i.e non-null), output for each compositum, not only the compositum polynomial pol, but a vector [pol,al1,al2,k] where al1 (resp. al2) is a root of pol1 (resp. pol2) expressed as a polynomial modulo pol, and a small integer k such that al2+k*al1 is the chosen root of polpolcyclo(n,{v=x}): n-th cyclotomic polynomial (in variable v)poldegree(x,{v}): degree of the polynomial or rational function x with respect to main variable if v is omitted, with respect to v otherwise. Return -1 if x = 0, and 0 if it's a non-zero scalarpoldisc(x,{v}): discriminant of the polynomial x, with respect to main variable if v is omitted, with respect to v otherwisepoldiscreduced(f): vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial fpolgalois(x): Galois group of the polynomial x (see manual for group coding)polhensellift(x, y, p, e): lift the factorization y of x modulo p to a factorization modulo p^e using Hensel lift. The factors in y must be pairwise relatively prime modulo ppolinterpolate(xa,{ya},{x},{&e}): polynomial interpolation at x according to data vectors xa, ya (ie return P such that P(xa[i]) = ya[i] for all i). If ya is omitter, return P such that P(i) = xa[i]. If present, e will contain an error estimate on the returned valuepolisirreducible(x): true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantpollead(x,{v}): leading coefficient of polynomial or series x, or x itself if x is a scalar. Error otherwise. With respect to the main variable of x if v is omitted, with respect to the variable v otherwisepollegendre(n,{v=x}): legendre polynomial of degree n (n C-integer), in variable vpolrecip(x): reciprocal polynomial of xpolred(x,{flag=0},{p}): reduction of the polynomial x (gives minimal polynomials only). Second and third args are optional. The following binary digits of flag are significant 1: partial reduction, 2: gives also elements. p, if present, contains the complete factorization matrix of the discriminantpolredabs(x,{flag=0}): a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 norm. flag is optional, whose binary digit mean 1: give the element whose characteristic polynomial is the given polynomial. 4: give all polynomials of minimal T2 norm (give only one of P(x) and P(-x))polredord(x): reduction of the polynomial x, staying in the same orderpolresultant(x,y,{v},{flag=0}): resultant of the polynomials x and y, with respect to the main variables of x and y if v is omitted, with respect to the variable v otherwise. flag is optional, and can be 0: default, assumes that the polynomials have exact entries (uses the subresultant algorithm), 1 for arbitrary polynomials, using Sylvester's matrix, or 2: using a Ducos's modified subresultant algorithmpolroots(x,{flag=0}): complex roots of the polynomial x. flag is optional, and can be 0: default, uses Schonhage's method modified by Gourdon, or 1: uses a modified Newton methodpolrootsmod(x,p,{flag=0}): roots mod p of the polynomial x. flag is optional, and can be 0: default, or 1: use a naive search, useful for small ppolrootspadic(x,p,r): p-adic roots of the polynomial x to precision rpolsturm(x,{a},{b}): number of real roots of the polynomial x in the interval]a,b] (which are respectively taken to be -oo or +oo when omitted)polsubcyclo(n,d,{v=x}): finds an equation (in variable v) for the d-th degree subfield of Q(zeta_n), where (Z/nZ)^* must be cyclicpolsylvestermatrix(x,y): forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowspolsym(x,n): vector of symmetric powers of the roots of x up to npoltchebi(n,{v=x}): Tchebitcheff polynomial of degree n (n C-integer), in variable vpoltschirnhaus(x): random Tschirnhausen transformation of the polynomial xpolylog(m,x,{flag=0}): m-th polylogarithm of x. flag is optional, and can be 0: default, 1: D_m~-modified m-th polylog of x, 2: D_m-modified m-th polylog of x, 3: P_m-modified m-th polylog of xpolzagier(n,m): Zagier's polynomials of index n,mprecision(x,{n}): change the precision of x to be n (n C-integer). If n is omitted, output real precision of object xprecprime(x): largest prime number<=x, 0 if x<=1prime(n): returns the n-th prime (n C-integer)primes(n): returns the vector of the first n primes (n C-integer)prod(X=a,b,expr,{x=1}): x times the product (X runs from a to b) of expressionprodeuler(X=a,b,expr): Euler product (X runs over the primes between a and b) of real or complex expressionprodinf(X=a,expr,{flag=0}): infinite product (X goes from a to infinity) of real or complex expression. flag can be 0 (default) or 1, in which case compute the product of the 1+expr insteadpsi(x): psi-function at xqfbclassno(x,{flag=0}): class number of discriminant x using Shanks's method by default. If (optional) flag is set to 1, use Euler productsqfbcompraw(x,y): Gaussian composition without reduction of the binary quadratic forms x and yqfbhclassno(x): Hurwitz-Kronecker class number of x>0qfbnucomp(x,y,l): composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputedqfbnupow(x,n): n-th power of primitive positive definite quadratic form x using nucomp and nuduplqfbpowraw(x,n): n-th power without reduction of the binary quadratic form xqfbprimeform(x,p): returns the prime form of discriminant x, whose first coefficient is pqfbred(x,{flag=0},{D},{isqrtD},{sqrtD}): reduction of the binary quadratic form x. All other args. are optional. D, isqrtD and sqrtD, if present, supply the values of the discriminant, floor(sqrt(D)) and sqrt(D) respectively. If D<0, its value is not used and all references to Shanks's distance hereafter are meaningless. flag can be any of 0: default, uses Shanks's distance function d; 1: use d, do a single reduction step; 2: do not use d; 3: do not use d, single reduction step. qfgaussred(x): square reduction of the (symmetric) matrix x (returns a square matrix whose i-th diagonal term is the coefficient of the i-th square in which the coefficient of the i-th variable is 1)qfjacobi(x): eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xqflll(x,{flag=0}): LLL reduction of the vectors forming the matrix x (gives the unimodular transformation matrix). flag is optional, and can be 0: default, 1: lllint algorithm for integer matrices, 2: lllintpartial algorithm for integer matrices, 3: lllrat for rational matrices, 4: lllkerim giving the kernel and the LLL reduced image, 5: lllkerimgen same but if the matrix has polynomial coefficients, 7: lll1, old version of qflll, 8: lllgen, same as qflll when the coefficients are polynomials, 9: lllint algorithm for integer matrices using contentqflllgram(x,{flag=0}): LLL reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix). flag is optional and can be 0: default,1: lllgramint algorithm for integer matrices, 4: lllgramkerim giving the kernel and the LLL reduced image, 5: lllgramkerimgen same when the matrix has polynomial coefficients, 7: lllgram1, old version of qflllgram, 8: lllgramgen, same as qflllgram when the coefficients are polynomialsqfminim(x,bound,maxnum,{flag=0}): number of vectors of square norm <= bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0. flag is optional, and can be 0: default; 1: returns the first minimal vector found (ignore maxnum); 2: as 0 but use Fincke-Pohst (valid for non integral quadratic forms)qfperfection(a): rank of matrix of xx~ for x minimal vectors of a gram matrix aqfsign(x): signature of the symmetric matrix xquadclassunit(D,{flag=0},{tech=[]}): compute the structure of the class group and the regulator of the quadratic field of discriminant D. If flag is non-null (and D>0), compute the narrow class group. See manual for the optional technical parametersquaddisc(x): discriminant of the quadratic field Q(sqrt(x))quadgen(x): standard generator of quadratic order of discriminant xquadhilbert(D,{flag=0}): relative equation for the Hilbert class field of the quadratic field of discriminant D (which can also be a bnf). If flag is a non-zero integer and D<0, list of [form,root(form)] (used for contructing subfields). If D<0, flag can also be a 2-component vector [p,q], where p,q are the prime numbers needed for Schertz's method. In that case, return 0 if [p,q] not suitable. If D>0 and flag is non-zero, try hard to find the best modulusquadpoly(D,{v=x}): quadratic polynomial corresponding to the discriminant D, in variable vquadray(D,f,{flag=0}): relative equation for the ray class field of conductor f for the quadratic field of discriminant D (which can also be a bnf). For D < 0, flag has the following meaning: if flag is an odd integer, output instead the vector of [ideal,corresponding root]. It can also be a two component vector [lambda,flag], where flag is as above and lambda is the technical element of bnf necessary for Schertz's method. In that case, return 0 if lambda is not suitable. For D > 0, if flag is non-zero, try hard to find the best modulusquadregulator(x): regulator of the real quadratic field of discriminant xquadunit(x): fundamental unit of the quadratic field of discriminant x where x must be positiverandom({N=2^31}): random integer between 0 and N-1real(x): real part of xremoveprimes({x=[]}): remove primes in the vector x (with at most 100 components) from the prime table. x can also be a single integer. List the current extra primes if x is omittedreorder({x=[]}): reorder the variables for output according to the vector x. If x is void or omitted, print the current list of variablesreturn({x=0}): return from current subroutine with result xrnfalgtobasis(rnf,x): relative version of nfalgtobasis, where rnf is a relative numberfieldrnfbasis(bnf,order): given an order as output by rnfpseudobasis or rnfsteinitz, gives either a basis of the order if it is free, or an n+1-element generating setrnfbasistoalg(rnf,x): relative version of nfbasistoalg, where rnf is a relative numberfieldrnfcharpoly(nf,T,alpha,{var=x}): characteristic polynomial of alpha over nf, where alpha belongs to the algebra defined by T over nf. Returns a polynomial in variable var (x by default)rnfconductor(bnf,polrel,{flag=0}): conductor of the Abelian extension of bnf defined by polrel. The result is [conductor,rayclassgroup,subgroup], where conductor is the conductor itself, rayclassgroup the structure of the corresponding full ray class group, and subgroup the HNF defining the norm group (Artin or Takagi group) on the given generators rayclassgroup[3]. If flag is non-zero, check (under GRH) that polrel indeed defines an Abelian extensionrnfdedekind(nf,T,pr): relative Dedekind criterion over nf, applied to the order defined by a root of irreducible polynomial T, modulo the prime ideal pr. Returns [flag,basis,val], where basis is a pseudo-basis of the enlarged order, flag is 1 iff this order is pr-maximal, and val is the valuation in pr of the order discriminantrnfdet(nf,order): given a pseudomatrix, compute its pseudodeterminantrnfdisc(nf,pol): given a pol with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfeltabstorel(rnf,x): transforms the element x from absolute to relative representationrnfeltdown(rnf,x): expresses x on the base field if possible; returns an error otherwisernfeltreltoabs(rnf,x): transforms the element x from relative to absolute representationrnfeltup(rnf,x): expresses x (belonging to the base field) on the relative fieldrnfequation(nf,pol,{flag=0}): given a pol with coefficients in nf, gives the absolute equation apol of the number field defined by pol. flag is optional, and can be 0: default, or non-zero, gives [apol,th], where th expresses the root of nf.pol in terms of the root of apolrnfhnfbasis(bnf,order): given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisernfidealabstorel(rnf,x): transforms the ideal x from absolute to relative representationrnfidealdown(rnf,x): finds the intersection of the ideal x with the base fieldrnfidealhnf(rnf,x): relative version of idealhnf, where rnf is a relative numberfieldrnfidealmul(rnf,x,y): relative version of idealmul, where rnf is a relative numberfieldrnfidealnormabs(rnf,x): absolute norm of the ideal xrnfidealnormrel(rnf,x): relative norm of the ideal xrnfidealreltoabs(rnf,x): transforms the ideal x from relative to absolute representationrnfidealtwoelt(rnf,x): relative version of idealtwoelement, where rnf is a relative numberfieldrnfidealup(rnf,x): lifts the ideal x (of the base field) to the relative fieldrnfinit(nf,pol): pol being a non constant irreducible polynomial defined over the number field nf, initializes a vector of data necessary for working in relative number fields (rnf functions). See manual for technical detailsrnfisfree(bnf,order): given an order as output by rnfpseudobasis or rnfsteinitz, outputs true (1) or false (0) according to whether the order is free or notrnfisnorm(bnf,ext,x,{flag=1}): Tries to tell whether x (in bnf) is the norm of some y (in ext). Returns a vector [a,b] where x=Norm(a)*b. Looks for a solution which is a S-integer, with S a list of places (in bnf) containing the ramified primes, generators of the class group of ext, as well as those primes dividing x. If ext/bnf is known to be Galois, set flag=0 (here x is a norm iff b=1). If flag is non zero add to S all the places above the primes: dividing flag if flag<0, less than flag if flag>0. The answer is guaranteed (i.e x norm iff b=1) under GRH, if S contains all primes less than 12.log(Ext)^2, where Ext is the normal closure of ext/bnfrnfkummer(bnr,subgroup,{deg=0}): bnr being as output by bnrinit, finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup. deg can be zero (default), or positive, and in this case the output is the list of all relative equations of degree deg for the given bnrrnflllgram(nf,pol,order): given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where neworder is a reduced order and U is the unimodular transformation matrixrnfnormgroup(bnr,polrel): norm group (or Artin or Takagi group) corresponding to the Abelian extension of bnr.bnf defined by polrel, where the module corresponding to bnr is assumed to be a multiple of the conductor. The result is the HNF defining the norm group on the given generators in bnr[5][3]rnfpolred(nf,pol): given a pol with coefficients in nf, finds a list of relative polynomials defining some subfields, hopefully simplerrnfpolredabs(nf,pol,{flag=0}): given a pol with coefficients in nf, finds a relative simpler polynomial defining the same field. flag is optional, 0 is default, 1 returns also the element whose characteristic polynomial is the given polynomial and 2 returns an absolute polynomialrnfpseudobasis(nf,pol): given a pol with coefficients in nf, gives a 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal order in HNF on the power basis, D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfsteinitz(nf,order): given an order as output by rnfpseudobasis, gives [A,I,D,d] where (A,I) is a pseudo basis where all the ideals except perhaps the last are trivialround(x,{&e}): take the nearest integer to all the coefficients of x. If e is present, do not take into account loss of integer part precision, and set e = error estimate in bitsserconvol(x,y): convolution (or Hadamard product) of two power seriesserlaplace(x): replaces the power series sum of a_n*x^n/n! by sum of a_n*x^n. For the reverse operation, use serconvol(x,exp(X))serreverse(x): reversion of the power series xsetintersect(x,y): intersection of the sets x and ysetisset(x): true(1) if x is a set (row vector with strictly increasing entries), false(0) if notsetminus(x,y): set of elements of x not belonging to ysetrand(n): reset the seed of the random number generator to nsetsearch(x,y,{flag=0}): looks if y belongs to the set x. If flag is 0 or omitted, returns 0 if it is not, otherwise returns the index j such that y==x[j]. If flag is non-zero, return 0 if y belongs to x, otherwise the index j where it should be insertedsetunion(x,y): union of the sets x and yshift(x,n): shift x left n bits if n>=0, right -n bits if n<0shiftmul(x,n): multiply x by 2^n (n>=0 or n<0)sigma(x,{k=1}): sum of the k-th powers of the divisors of x. k is optional and if omitted is assumed to be equal to 1sign(x): sign of x, of type integer, real or fractionsimplify(x): simplify the object x as much as possiblesin(x): sine of xsinh(x): hyperbolic sine of xsizebyte(x): number of bytes occupied by the complete tree of the object xsizedigit(x): maximum number of decimal digits minus one of (the coefficients of) xsolve(X=a,b,expr): real root of expression expr (X between a and b), where expr(a)*expr(b)<=0sqr(x): square of x. NOT identical to x*xsqrt(x): square root of xsqrtint(x): integer square root of x (x integer)sqrtn(x,n,{&z}): nth-root of x, n must be integer. If present, z is set to a suitable root of unity to recover all solutions. If it was not possible, z is set to zerosubgrouplist(bnr,{bound},{flag=0}): bnr being as output by bnrinit or a list of cyclic components of a finite Abelian group G, outputs the list of subgroups of G (of index bounded by bound, if not omitted), given as HNF left divisors of the SNF matrix corresponding to G. If flag=0 (default) and bnr is as output by bnrinit, gives only the subgroups for which the modulus is the conductorsubst(x,y,z): in expression x, replace the variable y by the expression zsum(X=a,b,expr,{x=0}): x plus the sum (X goes from a to b) of expression exprsumalt(X=a,expr,{flag=0}): Cohen-Villegas-Zagier's acceleration of alternating series expr, X starting at a. flag is optional, and can be 0: default, or 1: uses a slightly different method using Zagier's polynomialssumdiv(n,X,expr): sum of expression expr, X running over the divisors of nsuminf(X=a,expr): infinite sum (X goes from a to infinity) of real or complex expression exprsumpos(X=a,expr,{flag=0}): sum of positive series expr, the formal variable X starting at a. flag is optional, and can be 0: default, or 1: uses a slightly different method using Zagier's polynomialstan(x): tangent of xtanh(x): hyperbolic tangent of xtaylor(x,y): taylor expansion of x with respect to the main variable of yteichmuller(x): teichmuller character of p-adic number xtheta(q,z): Jacobi sine theta-functionthetanullk(q,k): k'th derivative at z=0 of theta(q,z)thue(tnf,a,{sol}): solve the equation P(x,y)=a, where tnf was created with thueinit(P), and sol, if present, contains the solutions of Norm(x)=a modulo units in the number field defined by P. If tnf was computed without assuming GRH (flag 1 in thueinit), the result is unconditionalthueinit(P,{flag=0}): initialize the tnf corresponding to P, that will be used to solve Thue equations P(x,y) = some-integer. If flag is non-zero, certify the result unconditionnaly. Otherwise, assume GRH (much faster of course)trace(x): trace of xtruncate(x,{&e}): truncation of x; when x is a power series,take away the O(X^). If e is present, do not take into account loss of integer part precision, and set e = error estimate in bitsuntil(a,seq): evaluate the expression sequence seq until a is nonzerovaluation(x,p): valuation of x with respect to pvariable(x): main variable of object x. Gives p for p-adic x, error for scalarsvecextract(x,y,{z}): extraction of the components of the matrix or vector x according to y and z. If z is omitted, y designs columns, otherwise y corresponds to rows and z to columns. y and z can be vectors (of indices), strings (indicating ranges as in "1..10") or masks (integers whose binary representation indicates the indices to extract, from left to right 1, 2, 4, 8, etc.)vecmax(x): maximum of the elements of the vector/matrix xvecmin(x): minimum of the elements of the vector/matrix xvecsort(x,{k},{flag=0}): sorts the vector of vectors (or matrix) x, according to the value of its k-th component if k is not omitted. Binary digits of flag (if present) mean: 1: indirect sorting, return the permutation instead of the permuted vector, 2: sort using ascending lexicographic ordervector(n,{X},{expr=0}): row vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0svectorv(n,{X},{expr=0}): column vector with n components of expression expr (X ranges from 1 to n). By default, fill with 0sweber(x,{flag=0}): One of Weber's f function of x. flag is optional, and can be 0: default, function f(x)=exp(-i*Pi/24)*eta((x+1)/2)/eta(x) such that (j=(f^24-16)^3/f^24), 1: function f1(x)=eta(x/2)/eta(x) such that (j=(f1^24+16)^3/f2^24), 2: function f2(x)=sqrt(2)*eta(2*x)/eta(x) such that (j=(f2^24+16)^3/f2^24)while(a,seq): while a is nonzero evaluate the expression sequence seq. Otherwise 0zeta(s): Riemann zeta function at szetak(nfz,s,{flag=0}): Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by zetakinit (NOT by nfinit) flag is optional, and can be 0: default, compute zetak, or non-zero: compute the lambdak function, i.e. with the gamma factorszetakinit(x): compute number field information necessary to use zetak, where x is an irreducible polynomialznlog(x,g): g as output by znprimroot (modulo a prime). Return smallest positive n such that g^n = xznorder(x): order of the integermod x in (Z/nZ)*znprimroot(n): returns a primitive root of n when it existsznstar(n): 3-component vector v, giving the structure of (Z/nZ)^*. v[1] is the order (i.e. eulerphi(n)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generators  $(,048<@DHLPTX\`dhlptx|  $(,048<@DHLPT~X}\|`{dzhylxpwtvxu|tsrqponmlkjihgfedcba`_^]\[ZYXWVUTSRQ PONML K$J(I,H0G4F8E<D@CDBHAL@P?T>X=\<`;d:h9l8p7t6x5|43210/.-,+*)('&%$#"!      $ ( ,048<@DHLPTX\`dhlptx|  $(,048<@DHLPTX\`dhlptx|  $(,048<@DHLPT~X}\|`{dzhylxpwtvxu|tsrqponmlkjihgfedcba`_^]\[ZYXWVUTSRQ PONML K$J(I,H0G4F8E<D@CDBHAL@P?T>X=\<`;d:h9l8p7t6x5|43210/.-,+*)('&%$#"!      .fileg..\..\..\..\OpenXM_contrib\pari-2.2\src\language\helpmsg.c@comp.id# .drectve(g.data ó$SG976 $SG975 $SG974 $SG973( $SG972$SG971$SG970$SG9698$SG968$SG967|$SG966$SG965$SG964$SG963`$SG962$SG961$SG960\$SG959$SG958T$SG957<$SG956T$SG9558$SG954$SG953$SG952$SG951P$SG950,$SG949$SG948L$SG947$SG946$SG945$SG944x$SG943,$SG942$SG941$SG940$SG939$SG938$SG937 $SG936$SG935$SG934`$SG933L$SG932$SG931$SG930d$SG9294$SG928$SG927$SG926$SG925$SG924P$SG923$SG922$SG921$SG920$SG919$SG918$SG917\$SG916T$SG9158$SG914$SG913$SG912$SG911h$SG910$SG9098$SG908T$SG907$SG906$SG905H$SG904$SG903$SG902$SG901($SG900$SG899|$SG898$SG897$SG896$SG895,$SG894$SG893t$SG892$SG891t$SG890($SG889`$SG888$SG887H$SG886$SG885H$SG884 $SG883$SG882$SG881$SG880|$SG879$SG878$SG877$SG876T$SG875$SG874@$SG873$SG872$SG871$SG870$SG869$SG868X$SG867,$SG866$SG865$SG864$SG863$SG862t$SG861$SG860|$SG859D$SG858$SG857X$SG856<$SG855|$SG854$SG853$SG852|$SG851L$SG850$SG849$SG848l$SG847$SG846\$SG845$SG844$SG843 $SG842$SG841 $SG840Ŀ$SG8390$SG838|$SG837$SG836$SG835@$SG834$SG833$SG832$SG831ȸ$SG830T$SG829H$SG828$SG827H$SG826ܵ$SG825\$SG824$SG823X$SG822$SG821ı$SG820$SG819L$SG818$SG817а$SG816$SG815d$SG814@$SG813 $SG812$SG811̮$SG8108$SG809$SG808$SG807D$SG806Ԭ$SG805$SG804$SG803X$SG802P$SG801$SG800$SG7990$SG798$SG797 $SG796L$SG795$SG794`$SG793($SG792С$SG791h$SG790$SG789$SG788$SG787$$SG786$SG785$SG784p$SG783$SG782$SG781$SG780 $SG779$SG778T$SG777$SG776$$SG775$SG774t$SG773($SG772$SG771ę$SG770$SG769$SG768`$SG767 $SG766d$SG765$SG764Ȗ$SG763|$SG762$SG761H$SG760$$SG759ȓ$SG758,$SG757$SG756$SG755$SG754Ȑ$SG753H$SG752$SG751`$SG750$SG749d$SG7480$SG747č$SG746$SG745܊$SG744$SG743$SG7428$SG741ԉ$SG740D$SG739$SG738`$SG737$SG736ȇ$SG735$SG734$$SG733$SG732t$SG731$SG730ԅ$SG729t$SG728@$SG727$SG726<$SG725$SG724$SG723l$SG722D$SG721$SG720$SG719<$SG718D$SG717$SG716~$SG7154~$SG714}$SG713}$SG712x}$SG711|$SG710d|$SG709{$SG708d{$SG707y$SG706x$SG705(x$SG704w$SG703v$SG702Xv$SG701u$SG700pu$SG699t$SG698s$SG6978r$SG696q$SG695@q$SG694tp$SG693 p$SG692Lo$SG691n$SG690n$SG689m$SG688k$SG687lk$SG6868k$SG685j$SG684j$SG683dj$SG6824j$SG681i$SG680h$SG679h$SG678h$SG677f$SG676f$SG675d$SG674b$SG673b$SG672a$SG671a$SG670`$SG6694`$SG668_$SG667_$SG666t_$SG665<_$SG664^$SG663L^$SG662]$SG661 ]$SG660\$SG659[$SG658T[$SG657[$SG656HZ$SG655pY$SG654TY$SG653Y$SG652X$SG6514X$SG650 X$SG649W$SG648DW$SG647U$SG646U$SG645T$SG644T$SG643hS$SG642R$SG641xR$SG640 R$SG639Q$SG638tQ$SG637Q$SG636P$SG635O$SG634O$SG633xN$SG632N$SG631@M$SG630xL$SG629L$SG628J$SG627xJ$SG626,J$SG625I$SG624H$SG623xH$SG622H$SG621G$SG620DG$SG619F$SG618F$SG617F$SG616E$SG615D$SG614D$SG613dD$SG612C$SG611C$SG610C$SG609`C$SG608LC$SG607XB$SG606PA$SG605@$SG604?$SG603?$SG602l?$SG601D?$SG600\>$SG599t=$SG598=$SG597,<$SG596;$SG595;$SG594t;$SG593;$SG5928$SG5917$SG5906$SG5895$SG5883$SG587 2$SG586(0$SG585/$SG584.$SG583.$SG582-$SG581(+$SG580($SG579p'$SG578&$SG5774&$SG576h%$SG575$$SG574$$SG573#$SG572"$SG571!$SG570$SG569$SG568($SG567$SG566$SG565H$SG564$SG563`$SG562$SG561$SG560$SG559L$SG558$SG557$SG556<$SG555$SG554p$SG553$SG552$SG551$SG550$SG549H$SG548$SG547$SG546$SG545t$SG544L$SG5430$SG542$SG541$SG540$SG539d$SG538 $SG537x $SG536X $SG535, $SG534 $SG533L $SG532 $SG531H $SG530 $SG529D $SG528 $SG5278 $SG526 $SG525$SG524 $SG523$SG522x$SG521\$SG520$_helpmessages_basic es.obj/ 1002273856 100666 60502 ` L @|;F.drectve( .bssi0.rdata@0@.data  @@.text\Zk P`.debug$FHB.rdata@@@.rdata@@@.rdata&@@@.rdata.@@@.rdata6@@@.rdata>@@@-defaultlib:LIBCMT -defaultlib:OLDNAMES stndrdth---- (type RETURN to continue) ----run-away string. Closing itrun-away comment. Closing itfailed read from file%020ld%c[0m%c[%d;%dm%c[%d;%d;%dm[+++]0.0.E%ldwr_float000000000E%ldgzero [SMALL %08lx ] [&=%08lx] ,CLONE%s(lg=%ld%s):%08lx chars:(%c,lgef=%ld):(%c,expo=%ld):%08lx (precp=%ld,valp=%ld):(%c,varn=%ld,lgef=%ld):(%c,varn=%ld,prec=%ld,valp=%ld):(lgef=%ld):%08lx int = pol = * mod = num = den = real = imag = * p : p^l : I : pol = real = imag = coef of degree %ld = %ld%s component = mat(%ld,%ld) = t_SMALLt_INTt_REALt_INTMODt_FRACt_FRACNt_COMPLEXt_PADICt_QUADt_POLMODt_POLt_SERt_RFRACt_RFRACNt_QFRt_QFIt_VECt_COLt_MATt_LISTt_STRt_VECSMALLunknown type %ld Top : %lx Bottom : %lx Current stack : %lx Used : %ld long words (%ld K) Available : %ld long words (%ld K) Occupation of the PARI stack : %6.2f percent %ld objects on heap occupy %ld long words %ld variable names used out of %d %08lx : %08lx NULL%ldMod(mod(wIO(^%ldO(^%ldQfb(qfr(qfi(List([])[;]matrix(0,%ld)matrix(0,%ld,j,k,0)Mat(mat(%08lx %ld##<%d>^%ld[;] ] ] %ldList(, ) mod /wI^%ld+ O( 1)^%ld + O( 1) / , , , } , [] ] [;] ] %08lx + - + - + {0}1} mod \overwI+ O(\cdot^{%ld}O(^{%ld}Qfb(qfr(qfi(, , , \pmatrix{ \cr} \pmatrix{ \cr} \pmatrix{ \cr \mbox{\pmatrix{ \cr this object uses debugging variablesTeX variable name too long_{%s}^{%ld} + - + - + \left(\right) \* \left(\right) \*I/O: opening file %s (code %d) closeI/O: closing file %s (code %d) could not open requested file %sI/O: opening file %s (mode %s) I/O: can't remove file %sI/O: removed file %s I/O: leaked file descriptor (%d): %sundefined environment variable: %sYou never gave me anything to read!%s/%sinputr%s.gprskipping directory %s.Z.gzgzip -d -c%s %saoutputwrite failedwrite failedwrite failedmalformed binary file (no GEN)malformed binary file (no name)malformed binary file (no GEN)setting %s unknown code in readobjread failedread failedread failedrabinary output%s is not a GP binary file%s not written for a %ld bit architectureunexpected endianness in %s%s written by an incompatible version of GP - -%c%s is set (%s), but is not writeable%s is 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1002273854 100666 7364 ` L>|;.drectve( .dataRr@0.bss 0-defaultlib:LIBCMT -defaultlib:OLDNAMES unexpected characterthis should be an integerincorrect type or length in matrix assignmentincorrect type in .too many parameters in user-defined function callunknown function or error in formal parametersvariable name expectedobsolete functionerror opening invalid flagWarning:Warning: increasing precWarning: failed toaccuracy problemsbug insorry,sorry, not yet available on this systemcollecting garbage inprecision too lowincorrect typeinconsistent dataimpossible assignment S-->Iimpossible assignment I-->Simpossible assignment I-->Iimpossible assignment R-->Simpossible assignment R-->Ioverflow in integer shiftoverflow in real shiftoverflow in truncationoverflow in S+Ioverflow in I+Ioverflow in I+Roverflow in R+Runderflow in R+Roverflow in I*Ioverflow in S*Roverflow in S*Ioverflow in R*Runderflow in R*Roverflow in I*R (R=0)division by zero in S/Sdivision by zero in S/Idivision by zero in S/Rdivision by zero in I/Sdivision by zero in I/Rdivision by zero in R/Sunderflow in R/Sdivision by zero in R/Idivision by zero in R/Runderflow in R/Roverflow in R/Runderflow in R/I (R=0)forbidden division R/R-->I or I/R-->I or R/I-->Idivision by zero in dvmdiizero modulus in modssdivision by zero in resssforbidden type in an arithmetic functionthird operand of type realthe PARI stack overflows !object too big, length can't fit in a codeworddegree overflowexponent overflowvaluation overflowunderflow or overflow in a R->dbl conversionimpossible concatenation in concatnon invertible matrix in gaussnot a square matrixnot linearly independent columns in supplunknown identifier valence, please reportbreak not allowednot an integer argument in an arithmetic functionnegative or zero argument in an arithmetic functionnegative argument in factorial functioninsufficient precision for p=2 in hildiscriminant not congruent to 0 or 1 mod 4primitive root does not exist in genernot enough precalculated primesimpossible inverse modulo: not a rational polynomialconstant polynomialnot a polynomialreducible polynomialzero polynomialnot a number field in some number field-related functionnot an ideal in an ideal-related functionnot a vector or incorrect vector length in ideallllred or minidealincorrect second argument in changevartoo many iterations for desired precision in integration routinenot a definite matrix in lllgramnot an integral matrix in lllgramintbad argument for an elliptic curve related functionpoint not on elliptic curveimpossibleforbiddendivision by zero in gdiv, gdivgs or ginva log/atan appears in the integration, PARI cannot handle thattrying to overwrite a universal objectnot enough memorysignificant pointers are lost in gerepile !!! (please report)not vectors in plothrawvectors not of the same length in plothrawincorrect type(s) or zero polynomial in rootpadic or factorpadicroot does not exist in rootpadicnonpositive precision in rootpadicinfinite precisionnegative exponentnon quadratic residue in gsqrtodd exponent in gsqrtnegative or zero integer argument in mpgammaq>=1 in thetawhat's going on ?~}| {zyxw v$u(t,s0r4q8p<o@nDmHlLkPjTiXf\e`ddchblap`t_x^|]\[ZYXWVUTSRQPONMLKJIHGFEDCBA@?>=<;: 98765 4$3(2,1004/8.<-@,D+H*L)P(T'X&\%`$d#h"l!p tx|   .fileg..\..\..\..\OpenXM_contrib\pari-2.2\src\language\errmsg.c@comp.id# .drectve(4.datar{bk$SG633$SG632|$SG631L$SG6304$SG629$SG628$SG627 $SG626 $SG625 $SG624` $SG6234 $SG622 $SG621 $SG620 $SG619 $SG618` $SG6174 $SG616( $SG615 $SG614 $SG613 $SG612 $SG611 $SG610< $SG609 $SG608 $SG607 $SG606h $SG605X $SG604@ $SG603, $SG602 $SG601 $SG600 $SG599 $SG598 $SG597l $SG596D $SG595 $SG594$SG593$SG592$SG591t$SG590H$SG5894$SG588$SG587$SG586$SG585$SG584$SG583$SG582X$SG581<$SG580 $SG579$SG578$SG577$SG576$SG575p$SG574X$SG573H$SG5724$SG571$SG570$SG569$SG568$SG567$SG566$SG565$SG564x$SG563`$SG562H$SG5614$SG560$$SG559$SG558$SG557$SG556$SG555$SG554$SG553$SG552$SG551$SG550p$SG549T$SG5488$SG547$SG546$SG545$SG544$SG543$SG542.bss $SG541$SG540$SG539x$SG538P$SG537H$SG536@$SG535,$SG534$SG533$SG532$SG531$SG530$SG529$SG528$SG527$SG526$SG525t$SG524@$SG523,$SG522$SG521$SG520_errmessagecompat.obj/ 1002273854 100666 138538 ` L>|;4.drectve( .data+,,@@.bss 0.text@v}~ P`.debug$FB HB.rdata,@@@-defaultlib:LIBCMT -defaultlib:OLDNAMES 2!     \_^`[YZa >   S>>  >>    ! 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OabsGpacosGpacoshGpaddellGGGaddprimesGadjGagmGGpakellGGalgdepGLpalgdep2GLLpalgtobasisGGanellGLapellGGapell2GGapprpadicGGargGpasinGpasinhGpassmatGatanGpatanhGpbasisGfbasis2GfbasistoalgGGbernrealLpbernvecLbestapprGGpbezoutGGbezoutresGGbigomegaGbilhellGGGpbinGLbinaryGbittestGGboundcfGLboundfactGLbuchcertifylGbuchfuGpbuchgenGD0.3,G,D0.3,G,D5,G,D1,G,D4,L,D3,L,D0,L,pbuchgenforcefuGD0.3,G,D0.3,G,D5,G,D1,G,D4,L,D3,L,D3,L,pbuchgenfuGD0.3,G,D0.3,G,D5,G,D1,G,D4,L,D3,L,D2,L,pbuchimagGD0.1,G,D0.1,G,D5,G,buchinitGD0.3,G,D0.3,G,D5,G,D1,G,D4,L,D3,L,D-1,L,pbuchinitforcefuGD0.3,G,D0.3,G,D5,G,D1,G,D4,L,D3,L,D-3,L,pbuchinitfuGD0.3,G,D0.3,G,D5,G,D1,G,D4,L,D3,L,D-2,L,pbuchnarrowGpbuchrayGGpbuchrayinitGGpbuchrayinitgenGGpbuchrealGD0,G,D0.1,G,D0.1,G,D5,G,pbytesizelGceilGcenterliftGcfGpcf2GGpchangevarGGcharGnchar1Gnchar2GnchellGGchineseGGchptellGGclassnoGclassno2GcoeffGLcompimagGGcompoGLcompositumGGcompositum2GGcomprealrawGGconcatGGconductorGD0,G,D0,G,D1,L,conductorofcharGGconjGconjvecGpcontentGconvolGGcoreGcore2GcorediscGcoredisc2GcosGpcoshGpcvtoiGfcycloLDndecodefactorGdecodemoduleGGdegreelGdenomGdeplinGpderivGndetGdet2GdetintGdiagonalGdilogGpdirdivGGdireulerV=GGIdirmulGGdirzetakGGdiscGdiscfGdiscf2GdiscrayabsGD0,G,D0,G,D0,L,discrayabscondGD0,G,D0,G,D2,L,discrayabslistGGdiscrayabslistarchGGLdiscrayabslistarchallGLdiscrayabslistlongGLdiscrayrelGD0,G,D0,G,D1,L,discrayrelcondGD0,G,D0,G,D3,L,divisorsGdivresGGdivsumGVIeigenGpeint1GperfcGpetaGpeulerpevalGexpGpextractGGfactLpfactcantorGGfactfqGGGfactmodGGfactorGfactoredbasisGGffactoreddiscfGGfactoredpolredGGpfactoredpolred2GGpfactornfGGfactorpadicGGLfactorpadic2GGLfactpolGLLfactpol2GLfiboLfloorGforvV=GGIfordivvGVIforprimevV=GGIforstepvV=GGGIforvecvV=GID0,L,fpnGLDnfracGgaloisGpgaloisapplyGGGgaloisconjGgaloisconj1GgaloisconjforceGgamhGpgammaGpgaussGGgaussmoduloGGGgaussmodulo2GGGgcdGGgetheapgetrandlgetstacklgettimelglobalredGgotos*hclassnoGhellGGphell2GGphermiteGhermite2GhermitehavasGhermitemodGGhermitemodidGGhermitepermGhessGhilblGGGhilbertLhilbplGGhvectorGVIhyperuGGGpiidealaddGGGidealaddmultoneGGidealaddoneGGGidealapprGGpidealapprfactGGidealchineseGGGidealcoprimeGGGidealdivGGGidealdivexactGGGidealfactorGGidealhermiteGGidealhermite2GGGidealintersectGGGidealinvGGidealinv2GGideallistGLideallistarchGGGideallistarchgenGGGideallistunitGLideallistunitarchGGGideallistunitarchgenGGGideallistunitgenGLideallistzstarGLideallistzstargenGLideallllredGGGpidealmulGGGidealmulredGGGpidealnormGGidealpowGGGidealpowredGGGpidealtwoeltGGidealtwoelt2GGGidealvallGGGidmatLifimagGimageGimage2GimagecomplGincgamGGpincgam1GGpincgam2GGpincgam3GGpincgam4GGGpindexrankGindsortGinitalgGpinitalgredGpinitalgred2GpinitellGpinitzetaGpintegGnintersectGGintgenV=GGIpintinfV=GGIpintnumV=GGIpintopenV=GGIpinverseimageGGisdiagonallGisfundGisideallGGisinclGGisinclfastGGisirreducibleGisisomGGisisomfastGGisoncurvelGGisprimeGD0,L,isprincipalGGisprincipalforceGGisprincipalgenGGisprincipalgenforceGGisprincipalrayGGisprincipalraygenGGispspGisqrtGissetlGissqfreeGissquareGisunitGGjacobiGpjbesselhGGpjellGpkaramulGGLkbesselGGpkbessel2GGpkerGkeriGkerintGkerint1Gkerint2GkroGGlabels*lambdakGGplaplaceGlcmGGlegendreLDnlengthlGlexlGGlexsortGliftGlindepGplindep2GLplllGplll1GplllgenGplllgramGplllgram1GplllgramgenGlllgramintGlllgramkerimGlllgramkerimgenGlllintGlllintpartialGlllkerimGlllkerimgenGlllratGlnGplngammaGplocalredGGlogGplogagmGplseriesellGGGGpmakebigbnfGpmatGmatextractGGGmathellGGpmatrixGGVVImatrixqzGGmatrixqz2Gmatrixqz3GmatsizeGmaxGGminGGminidealGGGpminimGGGminim2GGmodGGmodpGGmodreverseGmodulargcdGGmuGnewtonpolyGGnextprimeGnfdetintGGnfdivGGGnfdiveucGGGnfdivresGGGnfhermiteGGnfhermitemodGGGnfmodGGGnfmulGGGnfpowGGGnfreduceGGGnfsmithGGnfvallGGGnormGnorml2GnucompGGGnumdivGnumerGnupowGGoomegaGordellGGporderGorderellGGordredGppadicpreclGGpascalLDGperfGpermutationLGpermutation2numGpfGGpphiGpippnqnGpointellGGppolintGGGD&polredGppolred2GppolredabsGppolredabs2GppolredabsallGppolredabsfastpolredabsnoredGppolsymGLpolvarGpolyGnpolylogLGppolylogdLGppolylogdoldLGppolylogpLGppolyrevGnpolzagLLpowellGGGppowrealrawGLprecGLprecisionGprimeLprimedecGGprimesLprimrootGprincipalidealGGprincipalideleGGpprodGV=GGIprodeulerV=GGIpprodinfV=GIpprodinf1V=GIppsiGpqfiGGGqfrGGGGquaddiscGquadgenGquadpolyGrandomDGranklGrayclassnoGGrayclassnolistGGrealGrecipGredimagGredrealGredrealnodGGreduceddiscGregulaGpreorderGresultantGGresultant2GGreverseGrhorealGrhorealnodGGrndtoiGfrnfbasisGGrnfdiscfGGrnfequationGGrnfequation2GGrnfhermitebasisGGrnfisfreelGGrnflllgramGGGrnfpolredGGrnfpseudobasisGGrnfsteinitzGGrootmodGGrootmod2GGrootpadicGGLrootsGprootsof1GrootsoldGproundGrounderrorlGseriesGnsetGsetintersectGGsetminusGGsetrandlLsetsearchlGGD0,L,setunionGGshiftGLshiftmulGLsigmaGsigmakLGsignlGsignatGsignunitGsimplefactmodGGsimplifyGsinGpsinhGpsizelGsmallbasisGfsmallbuchinitGD0.3,G,D0.3,G,D5,G,D1,G,D4,L,D3,L,psmalldiscfGsmallfactGsmallinitellGpsmallpolredGpsmallpolred2GpsmithGsmith2GsmithcleanGsmithpolGsolveV=GGIpsortGsqrGsqredGsqrtGpsrgcdGGsturmlGsturmpartlGGGsubcycloGGDnsubellGGGpsubstGnGsumGV=GGIsumaltV=GIpsumalt2V=GIpsuminfV=GIpsumposV=GIpsumpos2V=GIpsupplementGsylvestermatrixGGtanGptanhGptaniyamaGtaylorGnPtchebiLDnteichGpthetaGGpthetanullkGLthreetotwoGGGGthreetotwo2GGGGtorsellGptraceGtransGtruncGtschirnhausGtwototwoGGGunitGuntilvaluationlGGvecGvecindexsortGveclexsortGvecmaxGvecminGvecsortGGvectorGVIvvectorGVIweipellGPwfGpwf2GpwhilezellGGpzetaGpzetakGGpzideallogGGGzidealstarGGzidealstarinitGGzidealstarinitgenGGznstarGO(a^b)=o(a^b)=p-adic or power series zero with precision given by babs(x)=absolute value (or modulus) of xacos(x)=inverse cosine of xacosh(x)=inverse hyperbolic cosine of xaddell(e,z1,z2)=sum of the points z1 and z2 on elliptic curve eaddprimes(x)=add primes in the vector x (with at most 20 components) to the prime tableadj(x)=adjoint matrix of xagm(x,y)=arithmetic-geometric mean of x and yakell(e,n)=computes the n-th Fourier coefficient of the L-function of the elliptic curve ealgdep(x,n)=algebraic relations up to degree n of xalgdep2(x,n,dec)=algebraic relations up to degree n of x where dec is as in lindep2algtobasis(nf,x)=transforms the algebraic number x into a column vector on the integral basis nf[7]anell(e,n)=computes the first n Fourier coefficients of the L-function of the elliptic curve e (n<32768)apell(e,p)=computes a_p for the elliptic curve e using Shanks-Mestre's methodapell2(e,p)=computes a_p for the elliptic curve e using Jacobi symbolsapprpadic(x,a)=p-adic roots of the polynomial x congruent to a mod parg(x)=argument of x,such that -pi0 in the wide sense. See manual for the other parameters (which can be omitted)bytesize(x)=number of bytes occupied by the complete tree of the object xceil(x)=ceiling of x=smallest integer>=xcenterlift(x)=centered lift of x. Same as lift except for integermodscf(x)=continued fraction expansion of x (x rational,real or rational function)cf2(b,x)=continued fraction expansion of x (x rational,real or rational function), where b is the vector of numerators of the continued fractionchangevar(x,y)=change variables of x according to the vector ychar(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using the comatrixchar1(x,y)=det(y*I-x)=characteristic polynomial of the matrix x using Lagrange interpolationchar2(x,y)=characteristic polynomial of the matrix x expressed with variable y, using the Hessenberg form. Can be much faster or much slower than char, depending on the base ringchell(x,y)=change data on elliptic curve according to y=[u,r,s,t]chinese(x,y)=x,y being integers modulo mx and my,finds z such that z is congruent to x mod mx and y mod mychptell(x,y)=change data on point or vector of points x on an elliptic curve according to y=[u,r,s,t]classno(x)=class number of discriminant xclassno2(x)=class number of discriminant xcoeff(x,s)=coefficient of degree s of x, or the s-th component for vectors or matrices (for which it is simpler to use x[])compimag(x,y)=Gaussian composition of the binary quadratic forms x and y of negative discriminantcompo(x,s)=the s'th component of the internal representation of x. For vectors or matrices, it is simpler to use x[]compositum(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2compositum2(pol1,pol2)=vector of all possible compositums of the number fields defined by the polynomials pol1 and pol2, with roots of pol1 and pol2 expressed on the compositum polynomialscomprealraw(x,y)=Gaussian composition without reduction of the binary quadratic forms x and y of positive discriminantconcat(x,y)=concatenation of x and yconductor(bnr,subgroup)=conductor of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupconductorofchar(bnr,chi)=conductor of the character chi on the ray class group bnrconj(x)=the algebraic conjugate of xconjvec(x)=conjugate vector of the algebraic number xcontent(x)=gcd of all the components of x, when this makes senseconvol(x,y)=convolution (or Hadamard product) of two power seriescore(n)=unique (positive of negative) squarefree integer d dividing n such that n/d is a squarecore2(n)=two-component row vector [d,f], where d is the unique squarefree integer dividing n such that n/d=f^2 is a squarecoredisc(n)=discriminant of the quadratic field Q(sqrt(n))coredisc2(n)=two-component row vector [d,f], where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f may be a half integercos(x)=cosine of xcosh(x)=hyperbolic cosine of xcvtoi(x)=truncation of x, without taking into account loss of integer part precisioncyclo(n)=n-th cyclotomic polynomialdecodefactor(fa)=given a factorisation fa, gives the factored object backdecodemodule(nf,fa)=given a coded module fa as in discrayabslist, gives the true moduledegree(x)=degree of the polynomial or rational function x. -1 if equal 0, 0 if non-zero scalardenom(x)=denominator of x (or lowest common denominator in case of an array)deplin(x)=finds a linear dependence between the columns of the matrix xderiv(x,y)=derivative of x with respect to the main variable of ydet(x)=determinant of the matrix xdet2(x)=determinant of the matrix x (better for integer entries)detint(x)=some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with hermitemoddiagonal(x)=creates the diagonal matrix whose diagonal entries are the entries of the vector xdilog(x)=dilogarithm of xdirdiv(x,y)=division of the Dirichlet series x by the Dir. series ydireuler(p=a,b,expr)=Dirichlet Euler product of expression expr from p=a to p=b, limited to b terms. Expr should be a polynomial or rational function in p and X, and X is understood to mean p^(-s)dirmul(x,y)=multiplication of the Dirichlet series x by the Dir. series ydirzetak(nf,b)=Dirichlet series of the Dedekind zeta function of the number field nf up to the bound b-1disc(x)=discriminant of the polynomial xdiscf(x)=discriminant of the number field defined by the polynomial x using round 4discf2(x)=discriminant of the number field defined by the polynomial x using round 2discrayabs(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdiscrayabscond(bnr,subgroup)=absolute [N,R1,discf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroup. Result is zero if fmodule is not the conductordiscrayabslist(bnf,listes)=if listes is a 2-component vector as output by ideallistunit or similar, gives list of corresponding discrayabsconddiscrayabslistarch(bnf,arch,bound)=gives list of discrayabscond of all modules up to norm bound with archimedean places arch, in a longvector formatdiscrayabslistarchall(bnf,bound)=gives list of discrayabscond of all modules up to norm bound with all possible archimedean places arch in reverse lexicographic order, in a longvector formatdiscrayabslistlong(bnf,bound)=gives list of discrayabscond of all modules up to norm bound without archimedean places, in a longvector formatdiscrayrel(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroupdiscrayrelcond(bnr,subgroup)=relative [N,R1,rnfdiscf] of the subfield of the ray class field bnr given by buchrayinit, defined by the HNF matrix subgroup. Result is zero if module is not the conductordivisors(x)=gives a vector formed by the divisors of x in increasing orderdivres(x,y)=euclidean division of x by y giving as a 2-dimensional column vector the quotient and the remainderdivsum(n,X,expr)=sum of expression expr, X running over the divisors of neigen(x)=eigenvectors of the matrix x given as columns of a matrixeint1(x)=exponential integral E1(x)erfc(x)=complementary error functioneta(x)=eta function without the q^(1/24)euler=euler()=euler's constant with current precisioneval(x)=evaluation of x, replacing variables by their valueexp(x)=exponential of xextract(x,y)=extraction of the components of the vector x according to the vector or mask y, from left to right (1, 2, 4, 8, ...for the first, second, third, fourth,...component)fact(x)=factorial of x (x C-integer), the result being given as a real numberfactcantor(x,p)=factorization mod p of the polynomial x using Cantor-Zassenhausfactfq(x,p,a)=factorization of the polynomial x in the finite field F_p[X]/a(X)F_p[X]factmod(x,p)=factorization mod p of the polynomial x using Berlekampfactor(x)=factorization of xfactoredbasis(x,p)=integral basis of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant of xfactoreddiscf(x,p)=discriminant of the maximal order defined by the polynomial x, where p is the matrix of the factorization of the discriminant of xfactoredpolred(x,p)=reduction of the polynomial x, where p is the matrix of the factorization of the discriminant of x (gives minimal polynomials only)factoredpolred2(x,p)=reduction of the polynomial x, where p is the matrix of the factorization of the discriminant of x (gives elements and minimal polynomials)factornf(x,t)=factorization of the polynomial x over the number field defined by the polynomial tfactorpadic(x,p,r)=p-adic factorization of the polynomial x to precision r, using the round 4 algorithmfactorpadic2(x,p,r)=p-adic factorization of the polynomial x to precision r, using Buchmann-Lenstrafactpol(x,l,hint)=factorization over Z of the polynomial x up to degree l (complete if l=0) using Hensel lift, knowing that the degree of each factor is a multiple of hintfactpol2(x,l)=factorization over Z of the polynomial x up to degree l (complete if l=0) using root findingfibo(x)=fibonacci number of index x (x C-integer)floor(x)=floor of x=largest integer<=xfor(X=a,b,seq)=the sequence is evaluated, X going from a up to bfordiv(n,X,seq)=the sequence is evaluated, X running over the divisors of nforprime(X=a,b,seq)=the sequence is evaluated, X running over the primes between a and bforstep(X=a,b,s,seq)=the sequence is evaluated, X going from a to b in steps of sforvec(x=v,seq)=v being a vector of two-component vectors of length n, the sequence is evaluated with x[i] going from v[i][1] to v[i][2] for i=n,..,1fpn(p,n)=monic irreducible polynomial of degree n over F_p[x]frac(x)=fractional part of x=x-floor(x)galois(x)=Galois group of the polynomial x (see manual for group coding)galoisapply(nf,aut,x)=Apply the Galois automorphism sigma (polynomial or polymod) to the object x (element or ideal) in the number field nfgaloisconj(nf)=list of conjugates of a root of the polynomial x=nf[1] in the same number field, using p-adics, LLL on integral basis (not always complete)galoisconj1(nf)=list of conjugates of a root of the polynomial x=nf[1] in the same number field nf, using complex numbers, LLL on integral basis (not always complete)galoisconjforce(nf)=list of conjugates of a root of the polynomial x=nf[1] in the Galois number field nf, using p-adics, LLL on integral basis. Guaranteed to be complete if the field is Galois, otherwise there is an infinite loopgamh(x)=gamma of x+1/2 (x integer)gamma(x)=gamma function at xgauss(a,b)=gaussian solution of ax=b (a matrix,b vector)gaussmodulo(M,D,Y)=one solution of system of congruences MX=Y mod Dgaussmodulo2(M,D,Y)=all solutions of system of congruences MX=Y mod Dgcd(x,y)=greatest common divisor of x and ygetheap()=2-component vector giving the current number of objects in the heap and the space they occupygetrand()=current value of random number seedgetstack()=current value of stack pointer avmagettime()=time (in milliseconds) since last call to gettimeglobalred(e)=e being an elliptic curve, returns [N,[u,r,s,t],c], where N is the conductor of e, [u,r,s,t] leads to the standard model for e, and c is the product of the local Tamagawa numbers c_pgoto(n)=THIS FUNCTION HAS BEEN SUPPRESSEDhclassno(x)=Hurwitz-Kronecker class number of x>0hell(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using theta-functionshell2(e,x)=canonical height of point x on elliptic curve E defined by the vector e computed using Tate's methodhermite(x)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, using a naive algorithmhermite2(x)=2-component vector [H,U] such that H is an (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, and U is a unimodular matrix such that xU=H, using Batut's algorithmhermitehavas(x)=3-component vector [H,U,P] such that H is an (upper triangular) Hermite normal form of x with extra zero columns, U is a unimodular matrix and P is a permutation of the rows such that P applied to xU gives H, using Havas's algorithmhermitemod(x,d)=(upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, where d is the non-zero determinant of this latticehermitemodid(x,d)=(upper triangular) Hermite normal form of x concatenated with d times the identity matrixhermiteperm(x)=3-component vector [H,U,P] such that H is an (upper triangular) Hermite normal form of x with extra zero columns, U is a unimodular matrix and P is a permutation of the rows such that P applied to xU gives H, using Batut's algorithmhess(x)=Hessenberg form of xhilb(x,y,p)=Hilbert symbol at p of x,y (integers or fractions)hilbert(n)=Hilbert matrix of order n (n C-integer)hilbp(x,y)=Hilbert symbol of x,y (where x or y is integermod or p-adic)hvector(n,X,expr)=row vector with n components of expression expr, the variable X ranging from 1 to nhyperu(a,b,x)=U-confluent hypergeometric functioni=i()=square root of -1idealadd(nf,x,y)=sum of two ideals x and y in the number field defined by nfidealaddone(nf,x,y)=when the sum of two ideals x and y in the number field K defined by nf is equal to Z_K, gives a two-component vector [a,b] such that a is in x, b is in y and a+b=1idealaddmultone(nf,list)=when the sum of the ideals in the number field K defined by nf and given in the vector list is equal to Z_K, gives a vector of elements of the corresponding ideals who sum to 1idealappr(nf,x)=x being a fractional ideal, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealapprfact(nf,x)=x being a prime ideal factorization with possibly zero or negative exponents, gives an element b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealchinese(nf,x,y)=x being a prime ideal factorization and y a vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0 for all other pidealcoprime(nf,x,y)=gives an element b in nf such that b.x is an integral ideal coprime to the integral ideal yidealdiv(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nfidealdivexact(nf,x,y)=quotient x/y of two ideals x and y in HNF in the number field nf when the quotient is known to be an integral idealidealfactor(nf,x)=factorization of the ideal x given in HNF into prime ideals in the number field nfidealhermite(nf,x)=hermite normal form of the ideal x in the number field nf, whatever form x may haveidealhermite2(nf,a,b)=hermite normal form of the ideal aZ_K+bZ_K in the number field K defined by nf, where a and b are elementsidealintersect(nf,x,y)=intersection of two ideals x and y in HNF in the number field defined by nfidealinv(nf,x)=inverse of the ideal x in the number field nf not using the differentidealinv2(nf,x)=inverse of the ideal x in the number field nf using the differentideallist(nf,bound)=vector of vectors of all ideals of norm<=bound in nfideallistarch(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, without generatorsideallistarchgen(nf,list,arch)=vector of vectors of all zidealstarinits of all modules in list with archimedean arch added, with generatorsideallistunit(bnf,bound)=2-component vector [L,U] where L is as ideallistzstar, and U is a vector of vector of zinternallogs of the units, without generatorsideallistunitarch(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitideallistunitarchgen(bnf,lists,arch)=adds the archimedean arch to the lists output by ideallistunitgenideallistunitgen(bnf,bound)=2-component vector [L,U] where L is as ideallistzstar, and U is a vector of vector of zinternallogs of the units, with generatorsideallistzstar(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, without generatorsideallistzstargen(nf,bound)=vector of vectors of all zidealstarinits of all ideals of norm<=bound, with generatorsideallllred(nf,x,vdir)=LLL reduction of the ideal x in the number field nf along direction vdir, in HNFidealmul(nf,x,y)=product of the two ideals x and y in the number field nfidealmulred(nf,x,y)=reduced product of the two ideals x and y in the number field nfidealnorm(nf,x)=norm of the ideal x in the number field nfidealpow(nf,x,n)=n-th power of the ideal x in HNF in the number field nfidealpowred(nf,x,n)=reduced n-th power of the ideal x in HNF in the number field nfidealtwoelt(nf,x)=two-element representation of an ideal x in the number field nfidealtwoelt2(nf,x,a)=two-element representation of an ideal x in the number field nf, with the first element equal to aidealval(nf,x,p)=valuation at p given in primedec format of the ideal x in the number field nfidmat(n)=identity matrix of order n (n C-integer)if(a,seq1,seq2)= if a is nonzero, seq1 is evaluated, otherwise seq2imag(x)=imaginary part of ximage(x)=basis of the image of the matrix ximage2(x)=basis of the image of the matrix ximagecompl(x)=vector of column indices not corresponding to the indices given by the function imageincgam(s,x)=incomplete gamma functionincgam1(s,x)=incomplete gamma function (for debugging only)incgam2(s,x)=incomplete gamma function (for debugging only)incgam3(s,x)=complementary incomplete gamma functionincgam4(s,x,y)=incomplete gamma function where y=gamma(s) is precomputedindexrank(x)=gives two extraction vectors (rows and columns) for the matrix x such that the exracted matrix is square of maximal rankindsort(x)=indirect sorting of the vector xinitalg(x)=x being a nonconstant irreducible polynomial, gives the vector: [x,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual),r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]initalgred(x)=x being a nonconstant irreducible polynomial, finds (using polred) a simpler polynomial pol defining the same number field, and gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see manual), r1+r2 first roots, integral basis, matrix of power basis in terms of integral basis, multiplication table of basis]initalgred2(P)=P being a nonconstant irreducible polynomial, gives a two-element vector [nf,mod(a,pol)], where nf is as output by initalgred and mod(a,pol) is a polymod equal to mod(x,P) and pol=nf[1]initell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j,[e1,e2,e3],w1,w2,eta1,eta2,q,area]initzeta(x)=compute number field information necessary to use zetak, where x is an irreducible polynomialinteg(x,y)=formal integration of x with respect to the main variable of yintersect(x,y)=intersection of the vector spaces whose bases are the columns of x and yintgen(X=a,b,s)=general numerical integration of s from a to b with respect to X, to be used after removing singularitiesintinf(X=a,b,s)=numerical integration of s from a to b with respect to X, where a or b can be plus or minus infinity (1.0e4000), but of same signintnum(X=a,b,s)=numerical integration of s from a to b with respect to Xintopen(X=a,b,s)=numerical integration of s from a to b with respect to X, where s has only limits at a or binverseimage(x,y)=an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwiseisdiagonal(x)=true(1) if x is a diagonal matrix, false(0) otherwiseisfund(x)=true(1) if x is a fundamental discriminant (including 1), false(0) if notisideal(nf,x)=true(1) if x is an ideal in the number field nf, false(0) if notisincl(x,y)=tests whether the number field defined by the polynomial x is isomorphic to a subfield of the one defined by y; 0 if not, otherwise all the isomorphismsisinclfast(nf1,nf2)=tests whether the number nf1 is isomorphic to a subfield of nf2 or not. If it gives a non-zero result, this proves that this is the case. However if it gives zero, nf1 may still be isomorphic to a subfield of nf2 so you have to use the much slower isincl to be sureisirreducible(x)=true(1) if x is an irreducible non-constant polynomial, false(0) if x is reducible or constantisisom(x,y)=tests whether the number field defined by the polynomial x is isomorphic to the one defined by y; 0 if not, otherwise all the isomorphismsisisomfast(nf1,nf2)=tests whether the number fields nf1 and nf2 are isomorphic or not. If it gives a non-zero result, this proves that they are isomorphic. However if it gives zero, nf1 and nf2 may still be isomorphic so you have to use the much slower isisom to be sureisoncurve(e,x)=true(1) if x is on elliptic curve e, false(0) if notisprime(x)=true(1) if x is a strong pseudoprime for 10 random bases, false(0) if notisprincipal(bnf,x)=bnf being output by buchinit, gives the vector of exponents on the class group generators of x. In particular x is principal if and only if the result is the zero vectorisprincipalforce(bnf,x)=same as isprincipal, except that the precision is doubled until the result is obtainedisprincipalgen(bnf,x)=bnf being output by buchinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vectorisprincipalgenforce(bnf,x)=same as isprincipalgen, except that the precision is doubled until the result is obtainedisprincipalray(bnf,x)=bnf being output by buchrayinit, gives the vector of exponents on the ray class group generators of x. In particular x is principal if and only if the result is the zero vectorisprincipalraygen(bnf,x)=bnf being output by buchrayinit, gives [v,alpha,bitaccuracy], where v is the vector of exponents on the class group generators and alpha is the generator of the resulting principal ideal. In particular x is principal if and only if v is the zero vectorispsp(x)=true(1) if x is a strong pseudoprime, false(0) if notisqrt(x)=integer square root of x (x integer)isset(x)=true(1) if x is a set (row vector with strictly increasing entries), false(0) if notissqfree(x)=true(1) if x is squarefree, false(0) if notissquare(x)=true(1) if x is a square, false(0) if notisunit(bnf,x)=bnf being output by buchinit, gives the vector of exponents of x on the fundamental units and the roots of unity if x is a unit, the empty vector otherwisejacobi(x)=eigenvalues and orthogonal matrix of eigenvectors of the real symmetric matrix xjbesselh(n,x)=J-bessel function of index n+1/2 and argument x, where n is a non-negative integerjell(x)=elliptic j invariant of xkaramul(x,y,k)=THIS FUNCTION HAS BEEN SUPPRESSEDkbessel(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)kbessel2(nu,x)=K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type)ker(x)=basis of the kernel of the matrix xkeri(x)=basis of the kernel of the matrix x with integer entrieskerint(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLkerint1(x)=LLL-reduced Z-basis of the kernel of the matrix x with rational entries using matrixqz3 and the HNFkerint2(x)=LLL-reduced Z-basis of the kernel of the matrix x with integral entries using a modified LLLkro(x,y)=kronecker symbol (x/y)label(n)=THIS FUNCTION HAS BEEN SUPPRESSEDlambdak(nfz,s)=Dedekind lambda function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)laplace(x)=replaces the power series sum of a_n*x^n/n! by sum of a_n*x^nlcm(x,y)=least common multiple of x and y=x*y/gcd(x,y)legendre(n)=legendre polynomial of degree n (n C-integer)length(x)=number of non code words in xlex(x,y)=compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x=n) matrix x into an integral matrix with gcd of maximal determinants equal to 1 if p is equal to 0, not divisible by p otherwisematrixqz2(x)=finds a basis of the intersection with Z^n of the lattice spanned by the columns of xmatrixqz3(x)=finds a basis of the intersection with Z^n of the Q-vector space spanned by the columns of xmatsize(x)=number of rows and columns of the vector/matrix x as a 2-vectormax(x,y)=maximum of x and ymin(x,y)=minimum of x and yminideal(nf,ix,vdir)=minimum of the ideal ix in the direction vdir in the number field nfminim(x,bound,maxnum)=number of vectors of square norm <= bound, maximum norm and list of vectors for the integral and definite quadratic form x; minimal non-zero vectors if bound=0minim2(x,bound)=looks for vectors of square norm <= bound, return the first one and its normmod(x,y)=creates the integer x modulo y on the PARI stackmodp(x,y)=creates the integer x modulo y as a permanent object (on the heap)modreverse(x)=reverse polymod of the polymod x, if it existsmodulargcd(x,y)=gcd of the polynomials x and y using the modular methodmu(x)=Moebius function of xnewtonpoly(x,p)=Newton polygon of polynomial x with respect to the prime pnextprime(x)=smallest prime number>=xnfdetint(nf,x)=multiple of the ideal determinant of the pseudo generating set xnfdiv(nf,a,b)=element a/b in nfnfdiveuc(nf,a,b)=gives algebraic integer q such that a-bq is smallnfdivres(nf,a,b)=gives [q,r] such that r=a-bq is smallnfhermite(nf,x)=if x=[A,I], gives a pseudo-basis of the module sum A_jI_jnfhermitemod(nf,x,detx)=if x=[A,I], and detx is a multiple of the ideal determinant of x, gives a pseudo-basis of the module sum A_jI_jnfmod(nf,a,b)=gives r such that r=a-bq is small with q algebraic integernfmul(nf,a,b)=element a.b in nfnfpow(nf,a,k)=element a^k in nfnfreduce(nf,a,id)=gives r such that a-r is the ideal id and r is smallnfsmith(nf,x)=if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of xnfval(nf,a,pr)=valuation of element a at the prime prnorm(x)=norm of xnorml2(x)=square of the L2-norm of the vector xnucomp(x,y,l)=composite of primitive positive definite quadratic forms x and y using nucomp and nudupl, where l=[|D/4|^(1/4)] is precomputednumdiv(x)=number of divisors of xnumer(x)=numerator of xnupow(x,n)=n-th power of primitive positive definite quadratic form x using nucomp and nuduplo(a^b)=O(a^b)=p-adic or power series zero with precision given by bomega(x)=number of unrepeated prime divisors of xordell(e,x)=y-coordinates corresponding to x-ordinate x on elliptic curve eorder(x)=order of the integermod x in (Z/nZ)*orderell(e,p)=order of the point p on the elliptic curve e over Q, 0 if non-torsionordred(x)=reduction of the polynomial x, staying in the same orderpadicprec(x,p)=absolute p-adic precision of object xpascal(n)=pascal triangle of order n (n C-integer)perf(a)=rank of matrix of xx~ for x minimal vectors of a gram matrix apermutation(n,k)=permutation number k (mod n!) of n letters (n C-integer)permutation2num(vect)=ordinal (between 1 and n!) of permutation vectpf(x,p)=returns the prime form whose first coefficient is p, of discriminant xphi(x)=Euler's totient function of xpi=pi()=the constant pi, with current precisionpnqn(x)=[p_n,p_{n-1};q_n,q_{n-1}] corresponding to the continued fraction xpointell(e,z)=coordinates of point on the curve e corresponding to the complex number zpolint(xa,ya,x)=polynomial interpolation at x according to data vectors xa, yapolred(x)=reduction of the polynomial x (gives minimal polynomials only)polred2(x)=reduction of the polynomial x (gives elements and minimal polynomials)polredabs(x)=a smallest generating polynomial of the number field for the T2 norm on the roots, with smallest index for the minimal T2 normpolredabs2(x)=gives [pol,a] where pol is as in polredabs, and alpha is the element whose characteristic polynomial is polpolredabsall(x)=complete list of the smallest generating polynomials of the number field for the T2 norm on the rootspolredabsfast(x)=a smallest generating polynomial of the number field for the T2 norm on the rootspolredabsnored(x)=a smallest generating polynomial of the number field for the T2 norm on the roots without initial polredpolsym(x,n)=vector of symmetric powers of the roots of x up to npolvar(x)=main variable of object x. Gives p for p-adic x, error for scalarspoly(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientpolylog(m,x)=m-th polylogarithm of xpolylogd(m,x)=D_m~-modified m-th polylog of xpolylogdold(m,x)=D_m-modified m-th polylog of xpolylogp(m,x)=P_m-modified m-th polylog of xpolyrev(x,v)=convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termpolzag(n,m)=Zagier's polynomials of index n,mpowell(e,x,n)=n times the point x on elliptic curve e (n in Z)powrealraw(x,n)=n-th power without reduction of the binary quadratic form x of positive discriminantprec(x,n)=change the precision of x to be n (n C-integer)precision(x)=real precision of object xprime(n)=returns the n-th prime (n C-integer)primedec(nf,p)=prime ideal decomposition of the prime number p in the number field nf as a vector of 5 component vectors [p,a,e,f,b] representing the prime ideals pZ_K+a.Z_K, e,f as usual, a as vector of components on the integral basis, b Lenstra's constantprimes(n)=returns the vector of the first n primes (n C-integer)primroot(n)=returns a primitive root of n when it existsprincipalideal(nf,x)=returns the principal ideal generated by the algebraic number x in the number field nfprincipalidele(nf,x)=returns the principal idele generated by the algebraic number x in the number field nfprod(x,X=a,b,expr)=x times the product (X runs from a to b) of expressionprodeuler(X=a,b,expr)=Euler product (X runs over the primes between a and b) of real or complex expressionprodinf(X=a,expr)=infinite product (X goes from a to infinity) of real or complex expressionprodinf1(X=a,expr)=infinite product (X goes from a to infinity) of real or complex 1+expressionpsi(x)=psi-function at xqfi(a,b,c)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c<0qfr(a,b,c,d)=binary quadratic form a*x^2+b*x*y+c*y^2 with b^2-4*a*c>0 and distance dquaddisc(x)=discriminant of the quadratic field Q(sqrt(x))quadgen(x)=standard generator of quadratic order of discriminant xquadpoly(x)=quadratic polynomial corresponding to the discriminant xrandom()=random integer between 0 and 2^31-1rank(x)=rank of the matrix xrayclassno(bnf,x)=ray class number of the module x for the big number field bnf. Faster than buchray if only the ray class number is wantedrayclassnolist(bnf,liste)=if listes is as output by idealisunit or similar, gives list of corresponding ray class numbersreal(x)=real part of xrecip(x)=reciprocal polynomial of xredimag(x)=reduction of the binary quadratic form x with D<0redreal(x)=reduction of the binary quadratic form x with D>0redrealnod(x,sq)=reduction of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]reduceddisc(f)=vector of elementary divisors of Z[a]/f'(a)Z[a], where a is a root of the polynomial fregula(x)=regulator of the real quadratic field of discriminant xreorder(x)=reorder the variables for output according to the vector xresultant(x,y)=resultant of the polynomials x and y with exact entriesresultant2(x,y)=resultant of the polynomials x and yreverse(x)=reversion of the power series xrhoreal(x)=single reduction step of the binary quadratic form x of positive discriminantrhorealnod(x,sq)=single reduction step of the binary quadratic form x with D>0 without distance function where sq=[sqrt D]rndtoi(x)=take the nearest integer to all the coefficients of x, without taking into account loss of integer part precisionrnfbasis(bnf,order)=given an order as output by rnfpseudobasis or rnfsteinitz, gives either a basis of the order if it is free, or an n+1-element generating setrnfdiscf(nf,pol)=given a pol with coefficients in nf, gives a 2-component vector [D,d], where D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfequation(nf,pol)=given a pol with coefficients in nf, gives the absolute equation of the number field defined by polrnfequation2(nf,pol)=given a pol with coefficients in nf, gives [apol,th], where apol is the absolute equation of the number field defined by pol and th expresses the root of nf[1] in terms of the root of apolrnfhermitebasis(bnf,order)=given an order as output by rnfpseudobasis, gives either a true HNF basis of the order if it exists, zero otherwisernfisfree(bnf,order)=given an order as output by rnfpseudobasis or rnfsteinitz, outputs true (1) or false (0) according to whether the order is free or notrnflllgram(nf,pol,order)=given a pol with coefficients in nf and an order as output by rnfpseudobasis or similar, gives [[neworder],U], where neworder is a reduced order and U is the unimodular transformation matrixrnfpolred(nf,pol)=given a pol with coefficients in nf, finds a list of polynomials defining some subfields, hopefully simplerrnfpseudobasis(nf,pol)=given a pol with coefficients in nf, gives a 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal order in HNF on the power basis, D is the relative ideal discriminant, and d is the relative discriminant in nf^*/nf*^2rnfsteinitz(nf,order)=given an order as output by rnfpseudobasis, gives [A,I,..] where (A,I) is a pseudo basis where all the ideals except perhaps the last are trivialrootmod(x,p)=roots mod p of the polynomial xrootmod2(x,p)=roots mod p of the polynomial x, when p is smallrootpadic(x,p,r)=p-adic roots of the polynomial x to precision rroots(x)=roots of the polynomial x using Schonhage's method modified by Gourdonrootsof1(nf)=number of roots of unity and primitive root of unity in the number field nfrootsold(x)=roots of the polynomial x using a modified Newton's methodround(x)=take the nearest integer to all the coefficients of xrounderror(x)=maximum error found in rounding xseries(x,v)=convert x (usually a vector) into a power series with variable v, starting with the constant coefficientset(x)=convert x into a set, i.e. a row vector with strictly increasing coefficientssetintersect(x,y)=intersection of the sets x and ysetminus(x,y)=set of elements of x not belonging to ysetrand(n)=reset the seed of the random number generator to nsetsearch(x,y)=looks if y belongs to the set x. Returns 0 if it is not, otherwise returns the index j such that y==x[j]setunion(x,y)=union of the sets x and yshift(x,n)=shift x left n bits if n>=0, right -n bits if n<0shiftmul(x,n)=multiply x by 2^n (n>=0 or n<0)sigma(x)=sum of the divisors of xsigmak(k,x)=sum of the k-th powers of the divisors of x (k C-integer)sign(x)=sign of x, of type integer, real or fractionsignat(x)=signature of the symmetric matrix xsignunit(bnf)=matrix of signs of the real embeddings of the system of fundamental units found by buchinitsimplefactmod(x,p)=same as factmod except that only the degrees of the irreducible factors are givensimplify(x)=simplify the object x as much as possiblesin(x)=sine of xsinh(x)=hyperbolic sine of xsize(x)=maximum number of decimal digits minus one of (the coefficients of) xsmallbasis(x)=integral basis of the field Q[a], where a is a root of the polynomial x where one assumes that no square of a prime>primelimit divides the discriminant of xsmallbuchinit(pol)=small buchinit, which can be converted to a big one using makebigbnfsmalldiscf(x)=discriminant of the number field defined by the polynomial x where one assumes that no square of a prime>primelimit divides the discriminant of xsmallfact(x)=partial factorization of the integer x (using only the stored primes)smallinitell(x)=x being the vector [a1,a2,a3,a4,a6], gives the vector: [a1,a2,a3,a4,a6,b2,b4,b6,b8,c4,c6,delta,j]smallpolred(x)=partial reduction of the polynomial x (gives minimal polynomials only)smallpolred2(x)=partial reduction of the polynomial x (gives elements and minimal polynomials)smith(x)=Smith normal form (i.e. elementary divisors) of the matrix x, expressed as a vectorsmith2(x)=gives a three element vector [u,v,d] where u and v are square unimodular matrices such that d=u*x*v=diagonal(smith(x))smithclean(z)=if z=[u,v,d] as output by smith2, removes from u,v,d the rows and columns corresponding to entries equal to 1 in dsmithpol(x)=Smith normal form (i.e. elementary divisors) of the matrix x with polynomial coefficients, expressed as a vectorsolve(X=a,b,expr)=real root of expression expr (X between a and b), where expr(a)*expr(b)<=0sort(x)=sort in ascending order of the vector xsqr(x)=square of x. NOT identical to x*xsqred(x)=square reduction of the (symmetric) matrix x ( returns a square matrix whose i-th diagonal term is the coefficient of the i-th square in which the coefficient of the i-th variable is 1)sqrt(x)=square root of xsrgcd(x,y)=polynomial gcd of x and y using the subresultant algorithmsturm(x)=number of real roots of the polynomial xsturmpart(x,a,b)=number of real roots of the polynomial x in the interval (a,b]subcyclo(p,d)=finds an equation for the d-th degree subfield of Q(zeta_p), where p must be a prime powersubell(e,z1,z2)=difference of the points z1 and z2 on elliptic curve esubst(x,y,z)=in expression x, replace the variable y by the expression zsum(x,X=a,b,expr)=x plus the sum (X goes from a to b) of expression exprsumalt(X=a,expr)=Villegas-Zagier's acceleration of alternating series expr, X starting at asumalt2(X=a,expr)=Cohen-Villegas-Zagier's acceleration of alternating series expr, X starting at asuminf(X=a,expr)=infinite sum (X goes from a to infinity) of real or complex expression exprsumpos(X=a,expr)=sum of positive series expr, the formal variable X starting at asumpos2(X=a,expr)=sum of positive series expr, the formal variable X starting at a, using Zagier's polynomialssupplement(x)=supplement the columns of the matrix x to an invertible matrixsylvestermatrix(x,y)=forms the sylvester matrix associated to the two polynomials x and y. Warning: the polynomial coefficients are in columns, not in rowstan(x)=tangent of xtanh(x)=hyperbolic tangent of xtaniyama(e)=modular parametrization of elliptic curve etaylor(x,y)=taylor expansion of x with respect to the main variable of ytchebi(n)=Tchebitcheff polynomial of degree n (n C-integer)teich(x)=teichmuller character of p-adic number xtheta(q,z)=Jacobi sine theta-functionthetanullk(q,k)=k'th derivative at z=0 of theta(q,z)threetotwo(nf,a,b,c)=returns a 3-component vector [d,e,U] such that U is a unimodular 3x3 matrix with algebraic integer coefficients such that [a,b,c]*U=[0,d,e]threetotwo2(nf,a,b,c)=returns a 3-component vector [d,e,U] such that U is a unimodular 3x3 matrix with algebraic integer coefficients such that [a,b,c]*U=[0,d,e]torsell(e)=torsion subgroup of elliptic curve e: order, structure, generatorstrace(x)=trace of xtrans(x)=x~=transpose of xtrunc(x)=truncation of x;when x is a power series,take away the O(X^)tschirnhaus(x)=random Tschirnhausen transformation of the polynomial xtwototwo(nf,a,b)=returns a 3-component vector [d,e,U] such that U is a unimodular 2x2 matrix with algebraic integer coefficients such that [a,b]*U=[d,e] and d,e are hopefully smallerunit(x)=fundamental unit of the quadratic field of discriminant x where x must be positiveuntil(a,seq)=evaluate the expression sequence seq until a is nonzerovaluation(x,p)=valuation of x with respect to pvec(x)=transforms the object x into a vector. Used mainly if x is a polynomial or a power seriesvecindexsort(x): indirect sorting of the vector xveclexsort(x): sort the elements of the vector x in ascending lexicographic ordervecmax(x)=maximum of the elements of the vector/matrix xvecmin(x)=minimum of the elements of the vector/matrix xvecsort(x,k)=sorts the vector of vector (or matrix) x according to the value of its k-th componentvector(n,X,expr)=row vector with n components of expression expr (X ranges from 1 to n)vvector(n,X,expr)=column vector with n components of expression expr (X ranges from 1 to n)weipell(e)=formal expansion in x=z of Weierstrass P functionwf(x)=Weber's f function of x (j=(f^24-16)^3/f^24)wf2(x)=Weber's f2 function of x (j=(f2^24+16)^3/f2^24)while(a,seq)= while a is nonzero evaluate the expression sequence seq. Otherwise 0zell(e,z)=In the complex case, lattice point corresponding to the point z on the elliptic curve ezeta(s)=Riemann zeta function at szetak(nfz,s)=Dedekind zeta function of the number field nfz at s, where nfz is the vector computed by initzeta (NOT by initalg)zideallog(nf,x,bid)=if bid is a big ideal as given by zidealstarinit or zidealstarinitgen , gives the vector of exponents on the generators bid[2][3] (even if these generators have not been computed)zidealstar(nf,I)=3-component vector v, giving the structure of (Z_K/I)^*. v[1] is the order (i.e. phi(I)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generatorszidealstarinit(nf,I)=6-component vector [I,v,fa,f2,U,V] where v is as in zidealstar without the generators, fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*zidealstarinitgen(nf,I)=6-component vector [I,v,fa,f2,U,V] where v is as in zidealstar fa is the prime ideal factorisation of I and f2, U and V are technical but essential to work in (Z_K/I)^*znstar(n)=3-component vector v, giving the structure of (Z/nZ)^*. v[1] is the order (i.e. phi(n)), v[2] is a vector of cyclic components, and v[3] is a vector giving the corresponding generatorsthis function has been suppressed (0@HP`hp (0@HP`hp (0@HP`hp (0@HP`hp (0@HP`hp (0~@}H|P{`zhpyxwvutsrqponml k(j0i@hHgPf`ehdpcba`_^]\[ZYXWVUT S(R0Q@PHOPN`MhLpKJIHGFEDCBA@?>=< ;(:09@8H7P6`5h4p3210/.-,+*)(' & % $ #( "0 !@ H P ` h p                 ( 0 @ H P ` h p                 ( 0 @ H P ` h p                 ( 0 @ H P ` h p                 ( 0 @ H P ` h p              (0@HP`hp (0@HP`hp (~0}@|H{Pz`yhxpwvutsrqponmlkj i(h0g@fHePd`chbpa`_^]\[ZYXWVUTSR Q(P0O@NHMPL`KhJpIHGFEDCBA@?>=< ;(:09@8H7P6`5h4p3210/.-,+*)('&%$ #("0!@ HP`hp  ( 0 @ H P`hp (0@HP`hp (0@HP`hp (0@HP`hp (0@HP`hp (0@HP`hp (0@~H}P|`{hzpyxwvutsrqponmlkj i(h0g@fHePd`chbpa`_^]\[ZYXWVUTSR Q(P0O@NHMPL`KhJpIHGFEDCBA@?>=<;: 9(;08@7H6P5`4h3p2130/.-,+*)('&%$ #("0!@ HP`hp    ( 0 @ H P ` h p             !!! !(!0!@!H!P!`!h!p!!!!!!!!!!!!!""" "("0"@"H"P"`"h"p"""""""""""""### #(#0#@#H#P#`#h#p#############$$$ $($0$@$H$P$`$h$p$$$$$$$$$$$$$%%% %(%0%@%H%P%`%h%p%%%%%%%%%%%%%&&& &(&~0&}@&|H&{P&z`&yh&xp&w&v&u&t&s&r&q&p&o&n&m&l&k'j'i'h 'g('f0'e@'dH'cP'b`'ah'`p'_'^']'\'['Z'Y'X'W'V'U'T'S(R(Q(P (O((N0(M@(LH(KP(J`(Ih(Hp(G(F(E(D(C(B(A(@(?(>(=(<(;):)9)8 )7()60)5@)4H)3P)2`)1h)0p)/).)-),)+)*)))()')&)%)$)#*"*!*  *(*0*@*H*P*`*h*p*********** + + +  + (+0+@+H+P+`+h+p+++++++++++++,,, ,(,0,@,H,P,`,h,p,,,,,,,,,,,,,--- -(-0-@-H-P-`-h-p-------------... .(.0.@.H.P.`.h.p............./// /(/0/@/H/P/`/h/p/////////////000 0(000@0H0P0`0h0p000000000000~0}1|1{1z 1y(1x01w@1vH1uP1t`1sh1rp1q1p1o1n1m1l1k1j1i1h1g1f1e2d2c2b 2a(2`02_@2^H2]P2\`2[h2Zp2Y2X2W2V2U2T2S2R22Q2P2O2N3M3L3K 3J(3I03H@3GH3FP3E`3Dh3Cp3B3A3@3?3>3=3<3;3:39383736454443 42(41040@4/H4.P4-`4,h4+p4*4)4(4'4&4%4$4#4"4!4 44555 5(505@5H5P5`5h5p5555555 5 5 5 5 55666 6(606@6H6P6`6h6p6666666666666777 7(707@7H7P7`7h7p7777777777777888 8(808@8H8P8`8h8p8888888888888999 9(909@9H9P9`9h9p9999999999999::: :(:0:@:H:P:`:h:p:::::::::::::;;; ;(;0;@;H;P;`;h;p;;;;;;;;;~;};|;{;z<y<x<w <v(<u0<t@<sH<rP<q`<ph<op<n<m<l<k<j<i<h<g<f<e<d<<c=b==a =`(=_0=^@=]H=\P=[`=Zh=Yp=X=W=V=U=T=S=R=Q==P=O=N=M>L >K(>J0>I@>HH>GP>F`>Eh>kp>D>C>>B>A>@>?>>>=><>;>:>9?8??7 ?6(?50?4@?3H?2P?1`?0h?/p?.?-?,?+?*?)?(?'?&?%?$@#@"@! @ (@0@@@H@P@`@h@p@@@@@@@@@@@ @ @ @ @ AAA AAAAA A$A(A,A0A4A8A,D=0D<4D;8D:,H=0H<4H;8H:|;8Y.drectve( .data:X@@.bss9@.textvp P`.debug$FvHB.rdataz@@@.rdata@@@.rdata@@@.textP P`.debug$F.HB-defaultlib:LIBCMT -defaultlib:OLDNAMES  d'@Bʚ;?a1a2a3a4a6areab2b4b6b8bnfc4c6clgpcodiffcycdiffdisceetaffufutugengroupjmodnfnoomegaordersppolregrootssigntatet2tutufuwzkzkstNULL[install] '%s' already there. Not replacednot a valid identifiercan't kill thatthis function uses a killed variableseqexprnot a proper member definitionhere (after ^)here (after !)incorrect vector or matrixhistory not availableexpected character: '%c' instead ofarray contexta 0x0 matrix has no elementsassignmentassignmentnot a suitable VECSMALL componentvariable on the left-hand side was affected during this function call. Check whether it is modified as a side effect therearray index (%ld) out of allowed range [none][1]%s[1-%ld]here (reading long)unfinished stringhere (argument reading)here (argument reading)not a variable:identifier (unknown code)can't derive thishere (in O()))test expressionstest expressionstest expressions%s already declared globalsymbol already in usehere (defining global var)here (reading function args)can't derive thislocal(user function %s: variable %Z declared twicehere (expanding string)here (print)global variable: exponent too largecan't pop gp variableno more variables availableidentifier already in use: %s%s already exists with incompatible valencerenaming a GP variable is forbiddenvariable number too bigunused charactersunused characters: %s; or ] expectedskipidentifier (unknown code)global variable not allowedegusing obsolete function %sefpbnfnfzkdiscpolmodsignt2diffcodiffrootsclgpclgpregray regulatorregfuray units.futuray torsion unitszkstnocycgengroupordersa1a2a3a4a6b2b4b6b8c4c6jomegacurve not defined over Retacurve not defined over Rareacurve not defined over Rtatecurve not defined over a p-adic fieldwcurve not defined over a p-adic fieldcan't modify a pre-defined member: unknown member functionpositive integer expectedpositive integer expectedunknown functiononly functions can be aliasedcan't replace an existing symbol by an aliaserrpiletypeergdiver2invmoderaccurerarcherthis trap keyword058P4Xp3x210//.0-8P,X!p+x*#)%('& 0%8P$Xp#x",! 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B0 -> %Z x2 -> %Z Checking for small solutions c5 = %Z c7 = %Z c10 = %Z c13 = %Z c6 = %Z c8 = %Z c11 = %Z c12 = %Z c14 = %Z c15 = %Z incorrect solutions of norm equationBaker -> %Z B0 -> %Z CF_First_Pass failed. Trying again with larger kappa CF_First_Pass successful !! 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Modulus = %Z and subgroup = %Z Cannot find a suitable modulus in FindModulus Trying modulus = %Z and subgroup = %Z CplxModuluscpl = %Z quadhilbertpolredsubfieldsmakescind (no polynomial found)Compute Wnmax in QuickPol: %ld zetavalues = %Z Checking the square-root of the Stark unit... polrelnum = %Z quickpolpolrelnumCompute %sstark (computation impossible)AllStarkpolrel = %Z Recpolnump = %ld InitGetRaydiff(chi) = %ZToo many coefficients (%Z) needed in GetST: computation impossibleNot enough precomputed primes (need all primes up to %ld)nmax = %ld and i0 = %ld Compute ann = %ld Compute S&TToo many coefficients (%ld) in QuadGetST: computation impossiblenmax = %ld Compute anCompute V1Compute V2bnrstarkthe ground field must be distinct from Qmain variable in bnrstark must not be xnot a totally real ground base field in bnrstarkincorrect subgroup in bnrstarknot a totally real class field in bnrstarknew precision: %ld the ground field must be distinct from QbnrL1incorrect subgroup in bnrL1incorrect subgroup in bnrL1no non-trivial character in bnrL1#  $c%}/|cHV\)x ct!)zZY D $ 0  " % -0/@3PU`YpZ[]_`kx 0@P`p 0@P`p    0!@#P)`*p:<=>GM)\(?? @Oޟ O?ƒPD#@ffffff?.fileg..\..\..\..\OpenXM_contrib\pari-2.2\src\modules\stark.c@comp.id# .drectve(A_:R.text$ ]  _gzero_gexp _gmul2n _gi_egalii _gdeux_gmul _denom _gdiv ! /_bot; 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numbers: %Z bounds computationprime ideal considered: %Z number of irreducible factors: %ld prime ideal chosen: %Z choice of the prime idealnew precision: %ld a suitable exponent is: %ld computation of Hcomputation of the p-adic factorizationcomputation of the factorsexponent: %ld nffactor[1]lllgram + base changenffactor[2]T2_matrix_powincorrect variables in rnfcharpolyrnfdedekindOޟ OOޟ O?L$VС++;v j=t5^Ð)1:@H.fileg..\..\..\..\OpenXM_contrib\pari-2.2\src\modules\nffactor.c@comp.id# .drectve(.bss,($.text`P:X_unifpol 2 @L _bot_avma.debug$F 弸V $L9769` n _lift z $L9774 $L9773_gmul $L9772$L11246$L11245,_gzero_gun _free @ `   _mpppcm _denom _content _gmod _factmod _gcmp1 _dvmdii  .data(=$SG10117 $SG10115_checknf      $SG10086L@ + 8 F $SG10355X _gdiv _shifti _gadd _modsi j _polunt _gsub _deriv _gpowgs _gcopy _polx $SG10229 $SG10222$SG10216x    $SG9909_gcmp0     _vals   _gsqr  0 _cmpsi  _nfrootsP  & _cmp_pol 0 : $SG10523pFR _gneg_i _ $SG105124$SG10506l" {  `# _srgcd _is_sqf# _addsi _discsr    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mpqs.obj/ 1002273843 100666 65614 ` L3|;.drectve( .data8-@@.bss90.textm 1 P`.rdata @@@.rdata(@@@.rdata0@@@.debug$F88 HB.rdatax@@@.rdata@@@.rdata@@@.rdata@@@.data@0.rdata@@@.rdata@@@.rdata@@@.rdata@@@.rdata@@@.rdata@@@.textP( P`.debug$FdtHB.data ~@0.rdata@@@-defaultlib:LIBCMT -defaultlib:OLDNAMES  !#%')+/3579;=ACEGIOSUY[]_aegikmq? @4@@*@@Q@ @? @5@@(@@Q@ @?@6@@(@@Q@@?@8@@(@@Q@@?@:@@(@@Q@@?0@=@@(@@Q@@?@@@@(@@N@ @?@A@@(@@N@ @?p@D@@(@@N@ @?p@N@@(@@I@$@? @T@@*@@I@$@?@@Y@@*@@D@$@?̰@Y@@*@@D@$@?@^@@*@@D@$@?@a@@,@@>@$@?@d@@,@@>@$@?@f@@,@@>@$@?p@i@@$@@>@$@Q?p@k@@$@@>@$@Q?d@n@@$@@>@$@Q?d@@p@@$@@>@$@(\?X@Pt@@$@@4@$@(\?X@0v@@$@@4@$@(\?L@pw@@$@@4@$@zG?L@y@@&@@4@$@zG?L@z@@&@@4@$@zG?L@0@@&@@4@$@zG?@@P@@&@@4@$@ ףp= ?@p@@&@@4@$@ ףp= ?@@@&@@4@$@ ףp= ?|@@@&@@4@$@ ףp= ?X@@@@&@@4@$@ ףp= ?X@@@&@ @$@$@ ףp= ?L@P@@&@ @$@$@ ףp= ?L@@@&@ @$@$@ ףp= ?L@@@(@"@$@$@ ףp= ?L@0@@(@"@$@$@ ףp= ?@@@(@"@$@$@ ףp= ?j@@@(@$@$@$@ ףp= ?@@@(@$@$@$@ ףp= ?L@@@(@$@$@$@?@@@(@$@$@$@?@p@@(@$@$@$@?j@@@(@&@$@$@?j@X@@(@&@$@$@333333?@@@*@&@$@$@333333?@@@*@&@$@$@333333?j@0@@*@&@$@$@333333?@\@ @,@(@$@$@333333?@@ @,@(@$@$@333333@@|@ @,@(@$@ @333333@A@ @,@(@$@ @333333@OAԷ@ @,@*@$@ @333333@A@ @,@*@$@ @@A,@ @,@*@$@ @@AX@ @,@*@$@ @@1A@ @,@*@$@ @@jA@ @,@*@$@ @333333@OAܾ@ @,@*@$@ @333333@OA@ @.@*@$@ @333333@@@ @.@*@ @@@@0@ @.@*@ @@@@0@"@0@*@ @@333333@A\@"@0@*@@@333333@zA@"@0@*@@@333333@A$@"@0@*@@@@A@"@1@*@@@@A\@"@1@*@@@@A@"@1@*@@@@jA$@"@1@*@@@333333@ A@"@2@*@@@333333@A@"@2@*@@@@`6A@@"@2@.@@@@`6A@"@2@.@@@@`6A@"@3@0@@@ @`6A@"@3@0@@@ @`6A@"@3@0@@@ @`6Av@"@3@0@@@ @`6Aj@"@4@1@@@ffffff @`6A^@"@4@1@@@ffffff @`6AX@"@4@1@@@ @>A@"@4@1@@@ @>A@@"@5@2@@@ @>A@"@5@2@@@ @>A@"@5@2@@@ @>AI@"@6@2@@@ @CAC@"@6@2@@@ffffff@CA@$@7@2@@@ffffff@CA1@$@7@2@@@ @`FA@$@8@2@@@ffffff@`FA@$@8@2@@@ffffff@JA@$@9@2@@@ffffff@JA^@$@9@2@@@ffffff@JA@$@:@2@@@333333@NA@$@:@2@@@333333@NA@$@;@2@@@333333@NA@$@;@2@@@333333@NAO@$@<@2@@@333333@NA@$@>@2@@@@ @ @ @ MPQS: number to factor N = %Z MPQS: number too big to be factored with MPQS, giving upMPQS: factoring number of %ld decimal digits manyseveralMPQS: the factorization of this number will take %s hoursMPQS: found multiplier %ld for N MPQS: kN = %Z MPQS: kN has %ld decimal digits MPQS: Gauss elimination will require more than 32MBy of memory (estimated memory needed: %4.1fMBy) MPQS: sieving interval = [%ld, %ld] MPQS: size of factor base = %ld MPQS: striving for %ld relations MPQS: first sorting at %ld%%, then every %3.1f%% / %3.1f%% MPQS: initial sieving index = %ld MPQS: creating factor base FB of size = %ld MPQS: largest prime in FB = %ld MPQS: bound for `large primes' = %ld MPQS: found factor = %ld whilst creating factor base MPQS: computing logarithm approximations for p_i in FB MPQS: computing sqrt(k*N) mod p_i MPQS: allocating arrays for self-initialization MPQS: number of prime factors in A is too smallMPQS: number of primes for A is too large, or FB too smallMPQS: index range of primes for A: [%ld, %ld] MPQS: coefficients A will be built from %ld primes each MPQS: starting main loop FRELFNEWLPRELLPNEWCOMBLPTMPwwwwMPQS: whilst trying to invert A4 mod kN, found factor = %Z MPQS: chose prime pattern 0x%lX for A MPQS: chose Q_%ld(x) = %Z x^2 - %Z x + C MPQS: chose Q_%ld(x) = %Z x^2 + %Z x + C sMPQS: found %lu candidate%s MPQS: passing the %3.1f%% checkpoint, time = %ld ms MPQS: passing the %3.1f%% checkpoint wr MPQS: split N whilst combining, time = %ld ms MPQS: found factor = %Z and combiningMPQS: done sorting%s, time = %ld ms MPQS: found %3.1f%% of the required relations MPQS: found %ld full relations MPQS: (%ld of these from partial relations) MPQS: %4.1f%% useless candidates MPQS: %4.1f%% of the iterations yielded no candidates MPQS: next checkpoint at %3.1f%% w MPQS: starting Gauss over F_2 on %ld distinct relations MPQS: time in Gauss and gcds = %ld ms MPQS: found factor = %Z MPQS: found factors = %Z and %Z MPQS: found %ld factors = , %Z%s MPQS: time in Gauss and gcds = %ld ms MPQS: no factors found. MPQS: restarting sieving ... MPQS: giving up. wrMQPS: short of space -- another buffer for sorting MQPS: line wrap -- another buffer for sorting MPQS: relations file truncated?! werror whilst writing to file %serror whilst writing to file %sMPQS: done sorting one file. rrcan't rename file %s to %sMPQS: renamed file %s to %s werror whilst writing to file %swerror whilst writing to file %serror whilst writing to file %serror whilst writing to file %serror whilst writing to file %serror whilst writing to file %swerror whilst writing to file %serror whilst writing to file %serror whilst writing to file %serror whilst writing to file %serror whilst writing to file %swerror whilst writing to file %serror whilst writing to file %serror whilst writing to file %serror whilst appending to file %serror whilst flushing file %sMPQS: precomputing auxiliary primes up to %ld MPQS: FB [-1,%ld...] Wait a second -- ,%ld] MPQS: last available index in FB is %ld MPQS: bin_index wraparound 0%s :%s 0%s @ %s :%s MPQS: combining {%ld @ %s : %s} * {%ld @ %s : %s} : 1 1 %ld %ld 0 == {%s} error whilst writing to file %ssMPQS: combined %ld full relation%s r\\ MATRIX READ BY MPQS FREL= \\ KERNEL COMPUTED BY MPQS KERNEL= MPQS: Gauss done: kernel has rank %ld, taking gcds... MPQS: no solutions found from linear system solver[1]: mpqs_solve_linear_systemMPQS: the combination of the relations is a nonsquare factoring (MPQS)[2]: mpqs_solve_linear_systemkNNMPQS: X^2 - Y^2 != 0 mod %s index i = %ld MPQS: wrong relation found after GausssMPQS: splitting N after %ld kernel vector%s MPQS: decomposed a square cube5th power7th powerMPQS: decomposed a %s MPQS: decomposed a square cube5th power7th powerMPQS: decomposed a %s MPQS: got two factors, looking for more... MPQS: resplitting a factor after %ld kernel vectors MPQS: decomposed a square cube5th power7th powerMPQS: decomposed a %s MPQS: decomposed a square cube5th power7th powerMPQS: decomposed a %s , looking for more...MPQS: got %ld factors%s [3]: mpqs_solve_linear_systemMPQS: wrapping up vector of %ld factors comp.unknown packaging %ld: %Z ^%ld (%s) [1, 0, 10; ] ftell error on full relations file ftell error on full relations filelongershorterMPQS: full relations file %s than expectedMPQS panicking can't seek full relations filefull relations file truncated?!U4S]VWjSD$4D$0 ɉ5uL$C~L$3+D$D$8D$<?D$$]8lj|$D$%yH@aD$SW \$8jVt$03Ƀ;t,F~ 3+ÃuD$0\$0E ;L$L$L$SQT$(5؋D$PR ɉ5u3D$ @~ Ǚ3+L$QPu4D$D$L$|$AL$ tD$0\$0T$E ;U|$D$0\$8AuD$0L$4D$8L$<|$(D$$=D$$dT$,D$(_^[]ÐT$S\$ t%CÉ [áUVqWu=P x+s j=t=~ G@ GމwWFuD$3-_0^][%~ jv} jދD$-_0^][Á3S$D$$$$$U-VWl$||ShSЃ3IRlr)hj -3_^][Ã=|VhFrTsPhj jS$|UhSUS\$TЃ3IR|ShVh$L$E7;D$ ;rD$ QL$3D$v$T$PD$F;rދL$RD$DD$ QT$D$TPh@$D$d;|*L$D$l@RPhD$,PhL$ -4 +D$x+;|$8v j=t=t j>7D$=t+Ǎ +;v j=t-t j>D$(ut T$H2ȡ+ +$;v j=t=t j>7;| hhhhhhhhRPhPP hQD$`hP$$@D$\L$D|$@t$x$$AJ;$u3F$D$$PD$xQL$tRT$lP$Q$VRT$8PD$LQL$4RT$HPD$xQ$RT$|P$4WQL$xSRPUQ$L$d$T;L$t5A%u y-$RQ=|^uPhE}.%?U @ED$T$<DŽ$$P߬$ $h2$DŽ$߬$ $h D$\P QjRP hQD$\T$D~YhPT$PL$ $QL$ RT$`PQRPW QD$@(#T$(RP jQRD$8$D$8$DŽ$$ ߬$$$$+ʋ$|$pۄ$ ڴ$H$|5|G $=D$3$D hRt$<|$8D$(D$,$LL$@D$DD$$u)$<0$D0PV#_^][(0Ë$<0T$8QhR $8WP} Qhj $D0tSV@_^][(0Í$8R苄$D0P$@ 3hQ$H$H |$,V;uSD$uhRD$D$L$QS}Rhj D$PV} Qhj D$1T$$RV}Phj Ëދ3$@0QhV D$t PT$(RD$ _^][(0Í$8PL$}D$%Ð 9tP@u+ÐVt$ W} k~kD$ P݆L$݆T$ =r L$$F!D$D$ l$ T$$݆|$(݆ L$,D$0I݆(L$4݆0D$8݆8T$< _ ^ÐD$T S\$PUVt$hW|$DE>ul$@J Vl$D9r"= h_^][ Ë\$P3Ʌv$D$Tӽ.t(*A;rl$@3D$Tu=P x+s j=t=~ G@ GމwǃD$\vs}PQOD$\uL$ ;tCpt%;}jL$$Nt׍+ы<8NuT$T%D$Xl$P+؉D$\$D$4(VQ=VPD$l ɉ=u3C~ ƙ3+L$\QVPPD$\u=Zx+s j=t=~ G@wGډWD$,VP ɉt T$,B~ ƙ3+u)j~ @@ @މpWPGu3+%~ j} jߋD$\PWL$(]QPD$l;t?wt%;} jNtύ+ˋNuD$TL$ D$PHD$Pl$@T$XD$\v"ʍ}qRPOD$\uL$ \$$;tCpt%;}jL$$Nt׍+Ӌ<8Nul$T-C%DutQQDPPS;ÉD$\t?pt%;} jNtύ+ˋNuL$ -jQT$8D$\ AL$Pr}|$dr+I|$L$D$\>-WPD$ ɉ-u3D$@~ Ǚ3+QPT$2D$HD$uT$@D$TңD$LL$X+ȉD$@L$LT$XD$P;D$0L$d+ȍpL$L$LT$@L$PR-D$d>D$ WP ɉ-u3D$@~ Ǚ3+L$@CDD$TD$P;zL$@D$XHL$@D$X>\$$|$P]D$4L$0+ȉL$LL$d+ȍxL$@L$8+ȉL$TL$<+ȉL$4L$PIL$d\$$T$LD$H-4:3VST$` ɉ-u3C~ ƙ3+++}֋D$@8PRȡ;r 3VQ T$`ŋڙ}֋D$@8PRȡ;r 3VQ T$`}ދD$T҉8}L$4D$d49HD$dD$,L$ -T$\RPjQP u.T$(D$\RPPUD$l$_^][ ËD$\\$h;Ët?pt%;} jNtύ+ˋNu-3D$@u @tD$@L$X-jRȍC+ƒL$\<uUA\$$y;D$Xu D$\M%? 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A.Y.a.j.v. ...E /R/SV/R]/Sn/Ry/S/r/+/b/+0?[0l00$000 0001  1l1}11$111 12222 G2 [2o2 2222 222 2233 3.3 43L3T3]3i3 r33 33333 33 34'404<4 E4d4>444 4445 5]5v55D6Q6Z6 x6666 647X7777 7888 $8@8+K8+8>8R8S8 9l99i 9 &9:9B9K9Q9 s9z9\9:999+9::i:$:0:@:G:S:q: }::::: :|:9: :;!;*;6; ?;a; g;;;;; ;;Q;|; <?<T<+<:<<+L=|V=S_=e==S=c===E>>>{+>7>>x>)~>>R>>R>S>>> >???? 2?P8?qF?LQ?SZ?=? ????? ?@L @a@@@@@@@q@PAL ASA=AL"A*AL1A7AL>A0RApWA]AqiA{A AAAA AApAA`AAApBSBBp1BHRBMYBdBMkBEqBJBDB|BBBHBRBSB BBBBB CJCACHnCLuC0C`C?CC DkDD&DL-Dk8D$BDLIDhDMoD{DLDEDJDKDwDHDRDSDkD\DJD EEE'E3E @IbGI_YIzbI:iISpI_IPIqIcI=I_II_I`IcIIJJ%J.JpCJzIJqSJS\JQmJrtJbJzJqJLJSJ=J`JJ`JJ KK K,K 5KNK`\K_kKsK KKKK KK_KKsKK_ L` L &LDLLLULaL jL{LpL?L`L`L LLLLL LsLM_Ms#M)M_JMLRM[MeMsmMM%MLMM&M\MSM"MLMM!MN%NLN"N&+N\2NS8NANLINPNfNNN8NN| OSOO6O>DO7QORXOShOlqOvOiOOlOOiOOOOOzO)O=O\PSPlPPi'P7PLPqPkP PPPPP P PPPQQ Q Q-Q5Q>QDQ OQ UQiQqQzQQ QvQ>QqQkQhQzQqQbQ_QQ R RRR (R=R>ER_RR|XR[cRlRvR_RR RRRR RR_SvS|"S+S4Sz=SSFSQPS_hS_zSkSS SSSS SS_TT T_-T 3TGTOTXTgT yTTkT_TT TTTT TU_UPUq%Uc.U=8U_EULUQUvfUmU^U\UUUzUU:UzUUSUUUQUSUzVP VqVcV= Vq*VP2Vc;V=DVSJVTV_VShVSnV~V\VVSVSVVVzVV:VSVVzVSV:W:WWSWz+WS4WQ=WSCWzPWSYW:nWvW}WSWWzWW:WzWSW:WWW_ XPXqXc!X=*XQ1X_@XQX_jXPpXqxXcX=XSX_XX XXYYY %Y>YDYqY{YY YYYYY Z+ ZdZ+Z4Z cDckKcWcbckc|cccc+ccdadndwdddd@eY\elneseiee eeeee f;fwfff} D" ]UNnM5~~s[[z Z* M =  0@P`p 0@P`p 0@Pkummer for composite relative degreekummer when zeta not in K requires a specific subgroupmain variable in kummer must not be xStep 1 Step 2 Step reduction polredabs = %ZStep 3 rnfkummerStep 4 Step 5 %ld Step 6 Step 7 and 8 Step 9 Step 10 and 11 Step 12 Step 13 Step 14 and 15 Step 16 Step 17 Step 18 main variable in kummer must not be xbug5 in kummerlistalpha = %Z bug 6: no equation found in kummerequations = %Zbug 7: more than one equation found in kummerbug%d in kummernot a virtual unit in isvirtualunitnot an element of K in downtoKreducing beta = %Z beta reduced = %Z polrelbe = %Z you should not be here in rnfkummer !!difficult Kummer for ell>=7reducebetareduce modulo (Z_K^*)^l unit exponents = %Z .fileg..\..\..\..\OpenXM_contrib\pari-2.2\src\modules\kummer.c@comp.id# .drectve(/G.textfvA_ellrank _divis _avma.debug$F`&tj  _FpV_red _gmulsg .dataM)$SG11294T* .bssx4<DD$SG11266H$SG11260<_modii _mulsi P c0_listbid`$SG11228,p {8d_listmodt$SG11213 _gdiv $SG11193   _egalii $SG11145 $SG11138  $SG11132_gtrans $SG11127_vecwh_dvH_FpM_ker _gsub $SG11111$SG11105_gsubst _gun_idmat $SG11079 $SG11063 _uuT   l_gencyc4_rcP_cyc _nfz(_bnfz\' $SG11038_U_gadd _powgi _gmul 1 _poleval $SG11033? $SG11031_addsi K _g_gpowgs _gener _d _m_degKz$$SG11023V _lift _gneg_i _pollead _R` _cyclo m $SG11013_elly_raycyc $SG11003`_degKX_vnf@_polnf,_nfL$SG10994( _gzero_dvmdii $SG10991_isprime _polx_gcmp1 _gellp _det _gcmp0   _bot 0! _gpow @" __append" @# $SG10132$SG10136$SG10135_gcopy _gegal  $SG10116_zsigne  $SG10042_gdivise  _ginv 1 $SG9894`? 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Dj\uʽҽ۽ ;@_-g{ izʿۿ [$ 2Z7HaeN-/EYTyc 1N _"6>G Q]|dba^`?i+h2SNpkwg7?H T]lm4BZbk w  /7@ LU{' >D_gp |n q<FZbk w p  Zx 9AJ V_Wrz$  %-6 BKdjpN-\B`$=-WB|# @g0z7g6yx#>FO [d '% ;Orci hz=K?d owt00vut )1: @Lh]zo=g$-)<g{-'~  $6x !)5S}Z`- -2:C I-" *<MWks| 06RZc mx !(-h=\j   -#7?H Nx#4G Y^   (-I RW| !nu{-  7>JW  @JZbk w  v8C-U-]qy !2FNW ] e e 7 KPj-qw   #<-DX`i o- I \ -2@IPT _glvII - '" /4e<Bair ~\` e +4n-B '`H V[enu e !* 4@hT-bh| ".5BYipz` $(,04&  8<5 Ix-% Z2[CZ*o F3 V%To3 x58E M #@z*  E^BB  RP*S9 ~  106@7P9`<p?_acw} 0@P`p 0@P`p 0@P`p"%<@ADIK L0M@NPY`]pfgjlnors{| 0@P`galoisconj2polconjugate %ld: %Z conjugate %ld: %Z rootsvandermondeinverseGaloisConj:val1=%ld val2=%ld GaloisConj: Bound %Z MonomorphismLift: lift to prec %dmonomorphismlift()MonomorphismLift: trying early solution %Z MonomorphismLift: true early solution. MonomorphismLift: false early solution. GaloisConj: Solution too large, discard it. GaloisConj:I will try %Z permutations Combinatorics too hard : would need %Z tests! I will skip it,but it may induce galoisinit to loopGaloisConj:Testing %ZGaloisConj:I will try %Z permutations Combinatorics too hard : would need %Z tests! I'll skip it but you will get a partial result...%d%% testpermutation(%Z)GaloisConj:%d hop sur %Z iterations testpermutation(%Z)GaloisConj:%d hop sur %Z iterations GaloisConj:Entree Verifie Test GaloisConj:Sortie Verifie Test:1 M%d.%ZGaloisConj:Sortie Verifie Test:0 First permutation shorter than second in permapplyFixedField: LN[%d]=%Z FixedField: Computed degrees: %Z prime too small in fixedfieldFixedField: Sym: %Z FixedField: bad mod: %Z FixedField: Tested: %Z %d incorrect permutation in permtopolIndexPartial: discriminantIndexPartial: factorizationIndexPartial: factor %Z --> %Z : GaloisConj:splitorbite: %Z entering black magic computationGaloisAnalysis:non Galois for p=%ld GaloisAnalysis:Nbtest=%ld,p=%ld,o=%ld,n_o=%d,best p=%ld,ord=%ld,k=%ld Galois group almost certainly not weakly super solvableGaloisAnalysis:non Galois for p=%ld GaloisAnalysis:p=%ld l=%ld group=%ld deg=%ld ord=%ld galoisanalysis()A4GaloisConj:I will test %ld permutations A4GaloisConj: %ld hop sur %ld iterations A4GaloisConj: %ld hop sur %ld iterations A4GaloisConj:sigma=%Z A4GaloisConj: %ld hop sur %ld iterations A4GaloisConj:tau=%Z A4GaloisConj:orb=%Z A4GaloisConj:O=%Z A4GaloisConj:%ld hop sur %d iterations max S4GaloisConj:Computing isomorphisms %d:%Z S4GaloisConj:Testing %d/3:%d/4:%d/4:%d/4:%Z S4GaloisConj:sigma=%Z S4GaloisConj:pj=%Z S4GaloisConj:Testing %d/3:%d/2:%d/2:%d/4:%Z:%Z S4GaloisConj:Testing %d/8 %d:%d:%d frobenius powers4test()GaloisConj:denominator:%Z GaloisConj:Testing A4 first GaloisConj:Testing S4 first GaloisConj:Orbite:%Z GaloisConj:Frobenius:%Z GaloisConj:P=%Z GaloisConj:psi=%Z GaloisConj:Sp=%Z GaloisConj:Pmod=%Z GaloisConj:Tmod=%Z GaloisConj:increase prec of p-adic roots of %ld. GaloisConj:Retour sur Terre:%Z GaloisConj:G[%d]=%Z d'ordre relatif %d GaloisConj:B=%Z GaloisConj:Paut=%Z GaloisConj:tau=%Z GaloisConj:g=%ld GaloisConj:w=%ld [%ld] sr=%ld dss=%ld GaloisConj:Fini! GaloisConj:Entree Init Test GaloisConj:Sortie Init Test galoisconj _may_ hang up for this polynomialGaloisConj:next p=%ld GaloisConj:p=%ld deg=%ld fp=%ld Trying degre %d. Galoisconj:Subgroups list:%Z Best lift: %d galoisconj4polynomial not in Z[X] in galoisconj4non-monic polynomial in galoisconj4Second arg. must be integer in galoisconj4galoisborne()rootpadicfast()vandermondeinversemod()GaloisConj:%Z Calcul polynomesNumberOfConjugates:Nbtest=%ld,card=%ld,p=%ld NumberOfConjugates:card=%ld,p=%ld conjugates list may be incomplete in nfgaloisconjnfgaloisconjplease apply galoisinit firstNot a Galois field in a Galois related functiongaloisinit: field not Galois or Galois group not weakly super solvablegaloispermtopolGaloisFixedField:cosets=%Z GaloisFixedField:den=%Z mod=%Z galoisfixedfieldgaloisfixedfieldgaloisfixedfieldGaloisConj:increase prec of p-adic roots of %ld. ypriority of optional variable too high in galoisfixedfieldgaloisisabelianwrong argument in galoisisabelianSubCyclo:elements:%Z SubCyclo:testing %ld^%ld SubCyclo:%ld not found SubCyclo:new conductor:%ld SubCyclo:elements:%Z SubCyclo:conductor:%ld galoisubcyclogaloissubcyclo for huge conductornot a HNF matrix in galoissubcycloOptionnal parameter must be as output by znstar in galoissubcycloMatrix of wrong dimensions in galoissubcycloznconductor.subgroupcoset.Subcyclo: orbit=%Z Subcyclo: %ld orbits with %ld elements each Subcyclo: prime l=%Z Subcyclo: borne=%Z Subcyclo: val=%ld padicsqrtnlift.computing roots.computing new roots.computing products.wrong modulus in galoissubcyclogenerators must be prime to conductor in galoissubcyclowrong type in galoissubcyclo6?D$u?u/L$uu QPQPËL$uu PQQPÐ4\?[c[nZvY.fileg..\..\..\..\OpenXM_contrib\pari-2.2\src\basemath\galconj.c@comp.id# .drectve( l.textP +'9  _lift  ,8 _bot_nfroots B _gsubst _polx_checknf _avma.debug$FpW8CM0 ] _cmp_pol g .data#$SG9625s_gdivise _poleval _gdiv  _lindep2 _gmul _gun.rdata_roots $SG9601_gzero  __ftol  $SG9694$_gconj     0     1 _gsub F  Z d o` |  _addii $SG9876t $SG9874T_gmul2n _addsr _gmulsg _gpowgs _gcmp _gadd $SG9855@_bfffo _gclone  _logint _gdeux _gabs  $SG98408 .textd5_mpmul _mulrr _mulir _mulii .debug$F _modii  _FpX_red 0 $SG10055 $SG10053 _gcopy _FpX_sub  _FpX_neg  & 7 A _sqri N X _deriv h { _timer2 p $SG10004$SG10002 $SG9999 0 _egalii  $SG9955D_cmpii   0 _dvmdii _shifti   0 .bss$SG10296$SG10298_addsi _mulss ' _FpX_add 3 $SG10249_cmpsi ?_divis $SG10247t_mpfact L`$ \p% $SG10646$SG10644$SG10651$SG10649$SG10589$SG10583@$SG10581m0 $SG10500t$SG10498p$SG10491l$SG10489h$SG10485Dz $SG10467$1 $SG10328_Vmatrix2 _gmod 3 _muldiv04 P4 _divll _mulll  p6 7 p8 8 ; _det    _znstar ( > 4pA H _zeropol R \E o }E F H _FpV_red 0I I $SG11230$SG11226$SG11216L `M $SG11145($SG11153\$SG11150@  !@O 1O FP ] _deg1pol i sS $SG11406tT  $SG11365xV _isprime _gpow  _gcmp1 X $SG11512$SG11511_mppgcd _respm $SG11509_powgi $SG11487$SG11485_ZX_disc @[ \ $SG11600_factor ` $SG11725<$SG11764$SG11768D$SG11766  $SG11749$SG11746d_cgcd  _diffptr$SG11660i $L11837;p$SG11927d$SG11905P$SG119028$SG11898 $SG11896$SG11860$SG11858$SG11856$L11836Wo$L11835%o$L11834n$L11831n$L17535hx_gop2z _gaffect $SG11808Xx $SG12276H$SG12238$SG12212$SG12210$SG12206+ $SG12143_Fp_isom _factmod 8 FP $SG10141lT f _s4test` $SG11992|r }`  _FpX_mul  $SG12865t $SG12844L $SG128328 _gegal $SG12822$ _FpX_gcd $SG12816 $SG12808 $SG12805 $SG12753  $SG12751| $SG12726h $SG12725T $SG12724@ $SG12723, $SG12722 $SG12707$SG12680$SG12673$SG12667$SG12663  $SG10426 _gtrans $SG10402 `  $SG12612 $SG12609  $SG12515h $SG12457H $SG124524 $SG12408  _cgiv  $P 5г Dp $SG12987D $SG129454 $SG12939 $SG12937 Q $SG12935 $SG12932 $SG12917 $SG12914 $SG12906x `@ $SG13043 $SG13041X t p $L13093 $SG13094 $L13088$L13085$L13077$SG13082 $L13067$L19783|  $SG13127 $SG13118 @ $SG13140@  $SG13178   $SG13290 $SG13285   $SG13452P $SG13450L$SG13447$SG13411$SG13401 $SG13377   p $SG13532$SG13505. =  MP $SG13688@$SG13680$SG13686($SG13682 $SG13669$SG13650Z _cyclo $SG13896$SG13894$SG13881$SG13873j _gener $SG13871$SG13868_binome $SG13866$SG13861X$SG13845D$SG138434$SG13841$$SG13823$SG13806$SG13795$SG13775h$SG13771Xz  $L13718$SG13740p$L13739$L13733$SG137328$L13726?$SG13724$L13721$L21313$$L213128_galoisconj_gerepileupto_dowin32ctrlc_win32ctrlc_pari_err_dummycopy_galoisconj2pol_gen_sort_fprintferr_DEBUGLEVEL_gtopolyrev__real@8@3ffce730e7c779b7c000_gisirreducible__fltused_galoisconj2_precision_nf_get_r1_permidentity_vandermondeinverseprepold_derivpol_vandermondeinverseprep_divide_conquer_prod_vandermondeinverse_gerepile_poldivres_galoisborne_gunclone_forcecopy_ceil_safe_ZX_disc_all_disable_dbg_genmsgtimer_gentimer_makeLden_initlift_monomorphismratlift_msgtimer_gerepilemanysp_FpX_Fp_add_FpXQ_mul_FpX_FpXQV_compo_FpXV_red_FpXQ_powers_FpXQ_inv_FpX_FpXQ_compo_hensel_lift_accel_brent_kung_optpow_monoratlift_gmodulcp_polratlift_poltopermtest_FpX_eval_automorphismlift_monomorphismlift_intheadlong_matheadlong_polheadlong_frobeniusliftall_FpX_center_FpX_Fp_mul_hiremainder_alloue_ardoise_testpermutation_verifietest_centermod_permapply_padicisint_hnftogeneratorslist_powmodulo_sortvecsmall_uniqvecsmall_pari_compare_lg_hnftoelementslist_listsousgroupes_subgrouplist_gtovecsmall_permorbite_fixedfieldpolsigma_FpXQ_pow_gmodulsg_fixedfieldfactmod_FpXQ_minpoly_fixedfieldnewtonsumaut_fixedfieldnewtonsum_fixedfieldpol_debug_surmer_fixedfieldsympol_fixedfieldtests_fixedfieldsurmer_FpX_is_squarefree_FpV_roots_to_pol_fixedfieldtest_fixedfieldinclusion_vandermondeinversemod_FpX_divres_mpinvmod_galoisgrouptopol_permtopol_normalizepol_i_corediscpartial_auxdecomp_indexpartial_permcyclepow_splitorbite_galoisanalysis_FpX_nbroots_simplefactmod_a4galoisgen_s4galoisgen_Fp_inv_isom_galoisdolift_inittestlift_bezout_lift_fact_s4makelift_scalarpol_s4releveauto_FpX_chinese_coprime_galoisgen_rootpadicliftroots_inittest_freetest_galoisfindfrobenius_galoisfrobeniuslift_galoisfindgroups_galoisfrobeniustest_galoismakepsiab_galoismakepsi_galoisconj4_rootpadicfast_numberofconjugates_sturmpart_galoisconj0_isomborne_checkgal_galoisinit_galoispermtopol_galoiscosets_fixedfieldfactor_vectopol_galoisfixedfield_fetch_user_var_galoistestabelian_galoisisabelian_subgroupcoset_sousgroupeelem_znconductor_galoissubcyclo_padicsqrtnlift_gscycloconductor_lift_check_modulus galois.obj/ 1002273831 100666 53560 ` L'|; .drectve( .bss0.dataDH@0.textyRR~Y P`.debug$F ̳BHB-defaultlib:LIBCMT -defaultlib:OLDNAMES  /pari/dataGaloisbig (prec=%ld): reduced polynomial #1 = %Z discriminant = %Z EVENODD%s group galois in degree > 11too large precision in preci()Partitions of %ld: p(%ld) = %ld i = %ld: %ld Output of isin_%ld_G_H(%ld,%ld): %ld Reordering of the roots: Output of isin_%ld_G_H(%ld,%ld): not included. ( %d ) RESGP_DATA_DIR%s/%s%ld_%ld_%ld_%ldgalois files not available in this version, sorryopening %sread_objectincorrect value in bin() ----> Group # %ld/%ld: all integer roots are double roots Working with polynomial #%ld: more than %ld rational integer roots $$$$$ New prec = %ld degree too large in tschirn $$$$$ Tschirnhaus transformation of degree %ld: $$$$$ indefinite invariant polynomial in gpoly() there are %ld rational integer roots: there is 1 rational integer root: there is no rational integer root. number%2ld: , order %ld. testing roots reordering: *** Entering isin_%ld_G_H_(%ld,%ld) COSCOS D$L$UVWPQt8;~HjEu RF;|_^]Á<SUVW=p+t$s j=3;t5"D$$$$$$$$$$$ $$$$$$$$0$$$@$$$$$$$`$$$$$$$$$$ $$@$$4$8 DŽ$8D$@D$DDŽ$ D$HD$LD$PD$TD$X$DŽ$( D$\D$`6DŽ$,PDŽ$0DŽ$<DŽ$@@DŽ$DNDŽ$Hl$C υt j>[ ׉t j>% ljE;wL$t$A;ʉL$t$H^][_YÐVt$ W|$ j+j2VWt _$^+Åu_2^j0j1VWu_1^j$j0VWu_0^jj$VW_$^$ÐD$ V/Wt$|$ j(j,VWtNjj(VWfjj(VWtjjVW_^_(^j&j,VWtjj&VWuh_&^j#j,VWu<_,^Ët$|$ j.j/VWj#j/VWjj#VWtjjVW_$^jj#VWtjjVW_$^jj#VW>jj#VWtjjVWB_^jj#VWjjVWjjVWtjjVW_^jjVWtvjjVWjjVWu4jjVWtjjVW_$^_^jjVW_$^_^_#^_/^jj.VWthjjVWt_^jjVWt4jjVWtjjVW_$^_^_^_.^ÐD$ V-Wt$|$ j j'VWt`jj VWj j VWj j VWtjj VW_$^ _ ^jj'VWjj'VW_'^Ët$|$ j*j-VWj)j-VWj"j)VWj!j)VWjj)VW$jjVWt~j jVWtNjj VWt_^jj VWt_^jj VW_$^ j jVW_^jjVWtj jVWj_^jjVWt4j jVW:j jVWP_^jjVW_^jj)VWtFjjVWuzj jVWXj jVW_^_)^_-^j"j*VWtFjj"VWu`jj"VWtjjVW_$^_"^j!j*VWjj!VWj jVWtjj VW_$^ j jVWtNjj VWt_^jj VWt_^jj VW_$^ _^j j!VWtjj VW_$^ _!^_*^ÐSVW=j ء<+1s j=tjjjj5j1jjjjFjjjFjjjjF @FjjjjjjjjjjFjjjjjjF@FjjjjjjjjjjjF jjjjjF$@F(jjjjjF,jjj jjjjj jjjF0@F4jj jjjjjjjjjjF8jjjjjjF<HF@jjj j jjjjjjjjjFDjj jjjjFHLFLjjjjjjj jjjjFPFTjjjjj@FXjjjj jjjjjjj jjjjF\jjjjF`LFdjjj j jjjjjjjj jjjjjjFhHFljjj j jjjjjj jjjjFpjj j jjFtjjjXFxjj jjjjjjjjj jjjjF|jjj jjjjXjj jjjjjjj j j jjjjjj Djjj jjjjjj jjjjjjj j j jjjjjj `jjj j jjjjjjjj j j jjjjj Ljjj j jjjjjjj jjjjjjDjjjjj jjjjjjjjj j j j j jjjjjj`jjj j jjjjjjjjj j j jjjjjj Tjjjjj j j j jjjjjjjjjjjjj j jjjj$H$DhD$ D$ 3VPSQR=_^[uá2ÐSUVW|$+D$O+;v j=tWWj55= Ƌу +‰T$@++ʉl$;v j=tL$+;|ET$0+B;T$}=~zt$FPWWh;~USh;|MRhF;~hD$C;|l$=p+s j=tD$5Hu_^][ÐSVW|$H+Í+;v j=tL$;ȉ5~@;TT|;Q @;~_^[Ë\$;}߾;|#Ul$OVW4EP FO;~]_^[ÐS\$VW3+NС++;v j=t=|D$OPNu_^[ÐHSD$U3VWt=Ht)Ht-l$6$! 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Increase precisiontorsellnot an integral curve in ellrootnonot a nonnegative integer second arg in ellrootnoincorrect prime in ellrootno_internL$VС++;v j=t5^Ð) 1:@HSVW|$uA\$t%@+t j@sC_^[Ët$} FF@UW+t j@F úFN ]~t%;}Bސ7 g zG!"@??@*B.6@W|$u_á Vp+s j=t5~F@~^_F~^_Ð 4, 4=CqA.fileg..\..\..\..\OpenXM_contrib\pari-2.2\src\modules\elliptic.c@comp.id# .drectve(6 .text@T__checkpt  .debug$Fk`9  ` # . < J X_bot_avmad  _gdiv .data@mD<$SG9707_gcmp0 _gmulsg _gsub _gmul _gadd _gmul2n _gsqr sP $SG9728}   $SG9751P$SG9750L $SG9748H_polx$SG9746$ _ellRHS _ellLHS _ellLHS0 _initell@ p _gzero$SG9976 _gexp _gabs _gimag _glog _gi_mppi _gexpo _gopsg2 _ginv _gneg_i _gsqrt  _greal _gsigne _roots $SG9946\_egalii _RHSpol  _do_agmP  $SG9784 $SG9774   $SG9872_gegal _gdivgs _grando0 $SG9843_gdeux$SG9841@ _gcmp1 _modii _invcmp _gcmp _coordch _checkch _pointch _gcopy P _oncurve  _addell  _gneg ' _subell@ _invell _ordell  1 _mulii ? M _powell$ $SG10382W& _poleval _deriv $SG10356_gpowgs _gun_gsubst $SG10329p_opgs2 _gnorm $SG10326L_denom $SG10321 _discsr _mathellp* _bilhell@- $SG10536b. _zell`0 _gfloor _dbltor _ghalfl $SG10640 $SG10639$SG10638$SG10637$SG10636$SG10635v$SG10600 6 $SG10572,7 _gzeta  $SG10769l_mulsi _PiI2 $SG10748`$SG107464_get_tau; $SG10688x`< _gconj _addii _ground .textP <> .debug$F.text)lg_affsr _bfffo .debug$F? _elleta0@ _ellzetaA $SG10903_ceil .rdata  _gaffect .data & _rtodbl $SG10881__ftol E _trueeta $SG10991$SG10976$SG10940N )O $SG10847_weipellU $L11059V$L11057U$L11055U$L11052U$L15409 W_ellwp00W $SG11131($SG11104 $SG11097_apell2pY $SG1118808Y _divll G_mulll _kross __a_2\ _ellap0P^ T^ b _dvmdii _apell1_ l v _cmpii _chinois  _decomp _mulss $SG11514$SG11471 $SG11455d _newbloc  _shifti _addsi _gceil  _timer2 _addsell o _sqri r _powsell0u __fixw _apellpx _cmpsi _gdivise $SG11737_apell0`z _free $SG11686_qsort  .rdata  .rdata ڢ .rdata S<  + 5  @ _anell _krogs _divis $SG11774$SG11770_akellP S $SG11862_hellp _gsin _hell2 _hells _numer _hell0 ^ $SG12151<_ghell0  _divrs _mulsr _ggval _factor _mppgcd _ghell2 _ghell j $SG12233x_incgam _incgam4 _ggamma _mpfactr _gpow _gopgs2 .rdatav_divrr $SG12206H $SG12560  $L12386$SG12387$L12385$L12384$L12383>_ggcd $L12377A$L12376$$L12375$L12374$L12373$L17794Ԫ$SG12358$SG12366 0  .text0_stoi .debug$F_auxp _aux2 p   p _gdivent _cumule _cumulev@ _cumule1@ " _gtoser _precdl, $SG127946 $SG12985<$SG12976,$SG12973I S $SG12821L_ratroot e@ u $L13200$L13191i$L13181$L13174$L19698p_gprec_w $SG13148x _cgcd _diffptr_torspnt  __roundp $SG13027_grndtoi _tors     $SG13220_torsell  $SG13577$SG135690 $L20271~$L13446$L13444$L13434$L13431<$L13421$L13428$L13427$L13426z$L13424I$L20597|$L134192$L13410$L13416$L13415$L13414$L13413v$L20596l$L13400_$L134072$L13406$L13405$L13404$L13403$L20595X$L13383$L13396$L13395G$L133881$L13386$L20594H$L13381$L20273$L20593 _neron $L13334$L13261U$L13327$L13325$L13324$L13316$L13308~$L13300c$L13298:$L13290%$L13287$L13283$L13282$L20640($L13259$L13258$L13257$L20639` $L13502$L13509$L13505t$L20782$L13500a$L13493I$L13507$L13484 $L13488$L13487$L20781$L13482$L20642$L20780$L13510  p $SG13546 _pari_err_checkell_checkbell_checksell_smallinitell_gerepilecopy_dowin32ctrlc_win32ctrlc_smallinitell0_ellinit0_ellprint_delete_var_fprintferr_name_var_fetch_var_initell0_thetanullk_gen_sort_gprecision_padic_initell_kronecker_do_padic_agm_pointch0_precision_gerepile_d_ellLHS_carrecomplet_gerepileupto_gmodulss_CM_powell_bilhells_flusherr_DEBUGLEVEL__fltused_new_coords_elleisnum_gerepilemany_DEBUGMEM_getgamma_new_chunk_get_periods__real@8@4002910a3d70a3d70800?reel4@?1??gtodouble@@9@9_ellsigma_pointell_weipellnumall_apell2_intern_hiremainder_multi_invmod_mpinvmod_gunclone_centermod_gmodulcp_msgtimer_dummycopy_lift_intern0_gmodulsg_addsell_part2_gpmalloc__real@8@3ffe8000000000000000__real@8@3fff8000000000000000__real@8@40018000000000000000_addsell1_powssell1_compare_multiples_auxdecomp_ellheight0_lseriesell__real@8@4003b17217f81956c000_localreduction_localreduction_carac_not23_ellLHS0_i_localreduction_result_localreduction_carac_23_numroots3_numroots2_globalreduction_taniyama_orderell_torsellnagelllutz_divisors_ellintegralmodel_is_new_torsion_torselldoud_torsbound_smaller_x_absi_cmp_best_in_cycle_elltors0_ellrootno_all_ellrootno_2_pvaluation_ellrootno_3_ellrootno_not23_ellrootno_intern_ellrootnotrans3.obj/ 1002273824 100666 66363 ` L |;6.drectve( .textP \] P`.debug$F1HB.data@0.rdatay@@@.rdata@@@.data@0.rdata@@@.rdata@@@.texty P`.debug$FHB.textP P`.debug$F=MHB.rdataW@@@.rdata_@@@.rdatag@@@.rdatao@@@.rdataw@@@-defaultlib:LIBCMT -defaultlib:OLDNAMES SUV5h+Ws j=t|$-E +Í +;v j=tlj5%D$t j>ߍ u+ +;v j=tD$5t j>u_^][ÐD$ t-Hthj 3ËD$L$T$PQR ËD$L$T$PQR ÐD$Xu' uT$dL$`RQP XS\$dUVWu3ցut$t-ȡ++;t$(|$Dv j=t=t j>VS~D$l|$X5D$, ++;l$Lv j=tlj-%D$t j>ߍ ]\$T+ +t$`;v j=tD$5t j>ȍ+ʋ+‰L$;vj=L$t L$D$ tj>L$ȍ+ʋ+‰L$p;vj=L$tt L$pD$ tj>L$t +‹+ʉD$,;vj=D$0 t D$,L$tj>D$0D$+ʋ+‰L$,;vj=L$0t L$,D$ t j>D$,D$\$lSPjSD$0uu PhPh T$(T$X D$X +WD$\L$,T$(D$LJ+L$\D$DQRhD$89D$tL$HPSQPD$4ȍ+ʃL$0 +;v j=tD$|$0=t j>T$T<ub A+D$D$TۉYσL$Pt%@+t j@_GD$$jP D$ L$(WjPD$| D$L$ WQHPف?j щD$pPL$pD$lQPD$$? 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(`g $SG109902@i <i _cmprr Lk .rdata"Y"_modulusn .rdata#$2tw#q  s _addsr @t _gexp _gnorm _mulsr .rdata$t>$y $SG11238_gconj { `| _split_1|  p $SG11200  # .p .rdata%O9%.rdata&W&.rdata'%-yu' _gclone    0   P @    $L11978$L11974$L11975$L15932 )0 5 F@ _isconj _roots0P Q $SG12194$[_square_free_factorization_gdivexact_modulargcd_derivpol_dowin32ctrlc_win32ctrlc_pari_err_isrealappr_gerepileupto_roots_com_all_roots_fprintferr_DEBUGLEVEL__real@8@bfffb8aa3b295c17f000__fltused_myrealun_mygprecrc_a_posteriori_errors__real@8@3fff8000000000000000_root_error_mygprec_absolute_split_complete__real@8@bffd9a209a84fbcfd800_polrecip_polrecip_i__real@8@3ffff333333333333000?reel4@?1??gtodouble@@9@9_max_modulus__real@8@3ffeb17217f7d1cf7800__real@8@40008000000000000000__real@8@40018000000000000000__real@8@3fffb8aa3b295c17f000__real@8@bfff8000000000000000__real@8@3fffc000000000000000__real@8@3ffcaaaaaaaaaaaaa800__real@8@3ffe8000000000000000__real@8@4000c000000000000000_set_karasquare_limit_COOK_SQUARE_LIMIT_KARASQUARE_LIMIT_cook_square_gerepilecopy_karasquare_mysquare_gerepile_quickmulcc_gerepilemanyvec__real@8@4002a000000000000000__real@8@40048000000000000000__real@8@4003b17217f7d1cf7800__real@8@c00fc350000000000000_findpower_pol_to_gaussint_myshiftrc_mygfloor_homothetie2n_homothetie_gauss_myshiftic_valuation_isexactzero_lower_bound__real@8@3fff9999999999999800_gpmalloc__real@8@00000000000000000000__real@8@3ffbccccccccccccd000__real@8@3ffcccccccccccccd000__real@8@3ffd8000000000000000_min_modulus_dual_modulus__real@8@3fffe000000000000000__real@8@3fff9249249249249000__real@8@3ffee000000000000000_eval_rel_pol_homothetie_optimize_split_split_fromU_parameters_gerepilemany_DEBUGMEM__real@8@c000c8f5c28f5c28f800_mpsincos_refine_F__real@8@4001a000000000000000_poldivres_refine_H_scalepol_compute_radius_pre_modulus__real@8@3ffdaaaaaaaaaaaaa800__real@8@bffeb17217f7d1cf7800_polygone_newton_update_radius_conformal_mapping__real@8@4000aed916872b021000_conformal_pol_split_0_1_scalepol2n_shiftpol_addshiftpol_scalarpol_split_0_2__real@8@c0079600000000000000__real@8@4000a000000000000000__real@8@400c9c40000000000000_append_root_cauchy_bound_mygprec_special_mygprecrc_special_fix_roots_roots_to_pol_gunclone_initRUgen_isexactpol_isexactscalar_isvalidcoeff_isvalidpol_solve_exact_pol_tocomplex_rootsoldpolarit3.obj/ 1002273815 100666 98208 ` L|;]V.drectve( .data@0.text P`.debug$F MWHB.bss 0-defaultlib:LIBCMT -defaultlib:OLDNAMES   non invertible polynomial in FpXQ_invpowers is only [] or [1] in FpX_FpXQV_compoFpX_FpXQV_compo: %d FpXQ_mul [%d] [1]: FpXQ_pow[2]: FpXQ_powFpXQX_safegcdffsqrtlmodffsqrtnmodffsqrtnmod1/0 exponent in ffsqrtnmodffsqrtnmodFF l-Gen:next %Zbad degrees in Fp_intersect: %d,%d,%dmatrixpow%Z is not a prime in Fp_intersectZZ_%Z[%Z]/(%Z) is not a field in Fp_intersectZZ_%Z[%Z]/(%Z) is not a field in Fp_intersectFpM_kerPolynomials not irreducible in Fp_intersectmpsqrtnmodpows [P,Q]ffsqrtnPolynomials not irreducible in Fp_intersectFpM_invimagepol[frobenius]matrix cyclokernelZZ_%Z[%Z]/(%Z) is not a field in Fp_intersectdivision by zero in FpX_divresdivision by zero in FpX_divrespolint_triv2 (i = %ld)ZY_ZXY_resultant_all: LERS needs lambdaStarting with lambda = %ld bound for resultant coeffs: 2^%ld Degree list for ERS (trials: %ld) = %Z Final lambda = %ld resultant mod %ld (bound 2^%ld, stable=%ld)ZY_ZXY_resultantbound for resultant: 2^%ld resultant mod %ld (bound 2^%ld, stable = %d)ZX_resultantmodulargcdmodulargcd: trial division failedmodulargcdZX_invmodZX_invmodZX_invmod: mod %ld (bound 2^%ld)ZX_invmod: char 0 check failedZX_invmodD$ÐD$ÐSUVt$W-~|$ȡ++;\$v j=tt j>ׁl$t$ }u5U p+s j=t5~ F@ F߉~D$ 3HD$ u\$|$t j?@É{_^][ÐS\$ VW{ȡ+ +;v j=t5t j>ׁۉ~D$NӋJut j?@Ɖ~_^[QD$SUVpXWN\$}-+ō +;|$v j=t=t j>֋΁,\$G tv;D$L$+D$D$8UPRȡ;r 3SQE;||$VW_^][Yÿ;~%D$l$+H)W3҃G;Q||$VW_^][YÐD$S\$VpW~8u N~t j?K3ҁ _? 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"! 4 !,KQiqz"+9D^`o|`" !.MTbv "  '-AIRX =XalO('3<y)7e(*^y+,4=Kb-5)c$k7v7yC*4368>KSgox~54*~83?S|Xem_21 W+&<'Oaa0nu//6@=USOdpolsym of a negative npolsympolsymMultiLift: bad argsbuilding treelifting to prec %ldrelatively prime polynomials expectednot a polynomial in polhenselliftnot a factorization in polhenselliftnot a prime number in polhenselliftnot a positive exponent in polhenselliftnot an integral factorization in polhenselliftnot a correct factorization in polhenselliftpolhensellift: factors %Z and %Z are not coprimepolsym of a negative npolsympolsympolsym_genpolsyn: non-invertible leading coeff: %ZLLL_cmbf: %ld potential factors (tmax = %ld) LLL_cmbf: (a, b) = (%ld, %ld), expo(T) = %ld LLL_cmbf: increasing BitPerFactor = %ld LLL_cmbf [no factor]LLL_cmbf: 1 factor LLL_cmbfspecial_pivot output: %Z LLL_cmbf: checking factor %ld LLL_cmbf: %ld factors impossible substitution in gdeflategdeflates %3ld factor%s of degree %3ld %3ld factor of degree %3ld s...tried prime %3ld (%-3ld factor%s). Time = %ld squff: wrong numbers of factorssplitting mod p = %ldMignotte bound: %Z^%ld last factor still to be checked making it monic ### K = %d, %Z combinations .T* to find factor %Zremaining modular factor(s): %ld factpolfactpolpartial factorization is not meaningful herefactor for general polynomialsfactor of general polynomialfactor of general polynomialcan't factor %Zprod: remaining objects %ld not a factorisation in factorbackgisirreduciblegcdggggggbezoutcontentmissing case in gdivexacteuclidean division by zero (pseudorem)pseudorem dx = %ld >= %ldeuclidean division by zero (pseudorem)pseudodiv dx = %ld >= %ldsubresall, dr = %ldsubresallsubresextsubresext, dr = %ldsubresextbezoutpolbezoutpol, dr = %ldinexact computation in bezoutpolresultantducos, degpol Q = %ldnextSousResultant j = %ld/%ldsylvestermatrixnot the same variables in sylvestermatrixpolresultantsrgcdsrgcd: dr = %ld srgcdpolgcdnundiscsrreduceddiscsmithreduceddiscsmithpoldiscreducednon-monic polynomial in poldiscreducedsturmsturmnot a squarefree polynomial in sturmpolsturm, dr = %ldzero discriminant in quadpolyquadpolyginvmodnon-invertible polynomial in polinvmodnon-invertible polynomial in polinvmodnon-invertible polynomial in polinvmodpolinvmodnon-invertible polynomial in polinvmodnewtonpolypolfnfpolfnf: choosing k = %ld reducible modulus in factornfmatratliftpolratliftnfgcd: p=%d nfgcd].bRh s LhoM  !" *CGbRUU3a {e,  i Y1:]?w^^X9` ~:8/s= **P:1ja cc{L  O    #0$@&P,`-p/6<>?EHN]dh o0t@vP`p 0@P`p$:<@B D0F@HPI`KpN 0@P`p 0@P`p 0@P ` p ')+,.@rZ| ??9B.?9B.6@.fileg..\..\..\..\OpenXM_contrib\pari-2.2\src\basemath\polarit2.c@comp.id# .drectve(.text nE_polsym  _gneg _gdiv _gadd _gmul _gmulsg _gcmp1 _gzero _bot_avma.data VJ$SG9581 $SG9577( $SG9572.debug$Fq42` .bss ?_polcmp Q b l` y _addii _cmpii _modii _shifti `    $SG10042L $SG10031<_timer2 $SG10017(    $SG9883`    _gpowgs @ + _gun7 E _mulii _gneg_i P _gcopy b q  _FpX_neg _FpX_sub  _FpX_mul 0  $SG10214_FpX_gcd  $SG10201T_gcmp0  $SG10191$_lift $SG10175$SG10173_isprime $SG10167$SG10154 _dvmdii _gsub _polx_denom  _sqri _logint  _bfffo     $SG10565_invmod $SG10549$SG10543$SG10539$SG10534p  _gscal _ginv _sqscal $ /" > HP# $SG10942$SG10904$SG10902$SG10930$SG10922R $SG10919`$SG10899x_cmprr j y  _vconcat  _zerocol $SG10869H_gexpo _gmod .rdata_ceil _dbltor .rdata'w_gauss   $SG10831.rdata"^7_idmat _glog .rdata  _gaffect .data &, _rtodbl F __ftol P / [ eP3 _addsi q4 _egalii _poleval }6 _gsqr _gcmp _gsigne 7 8 `8 8 _zeropol 9 _cgcd  : $SG11291T$SG112760< < _squff2= _decomp _squff@ __col $SG11204$SG11202   $SG11183$SG11182$SG11181 $SG11171$SG11168d$SG11167`$SG11166% 0 > I T ` u_divis  _diffptr_min_degpH H I     $SG11061T$SG110540.rdata ( $SG11015M _grndtoi _gmul2n _mpsqrt  _cmbf0N $SG10433$SG10374$SG10411$SG10410$SG10409$SG10381$SG10359$SG10337h_binome )PZ _racine _factpol[ ; L W c $SG11396$SG11392_factor0`b m $SG11709x b _factorc $L11850jk$L11770Do$SG11935t_concat $L11926n $L11782n_gsubst _gi $SG11899T $SG118854   $L11849Ok_gnorm $L11805g _roots $L11789e_factmod $L11788e$L11787e$SG11786$L11785e$L17986o$L11780e$L11775_e_gred $L11774Te$L11773Ce$L17985`o$L17984|o_poltypeo $L18043w$L115014u$L11644Nt$L11624s$L11611os$L11606*s$L11599r$L18067x$L11587r$L11574qr$L11556q$L11545q$L11543|q$L11538Dq$L18066w$L11521p_gegal $L11514tp $L11512Rp$L18065w0x $SG11957y $SG12026_powgi z . ;{ J W { c m@{ y `{ { $SG12073_gcd0|  $SG12097_ggcd} $L12340$L12309 $SG12437$SG12427$L12425$L12406W$L12401_ggval $L12396$L12395$L18479l$L12390u$L12389H$L12372$L12348$L12245k$L12325X$L12308E$L123292$L12312'$L12326$L12307$L12302($L12289~$L18478T$L12284k$L12255b$L12246$L12217 _mpppcm $L12210_top$L12185$L12184$L184774$SG12387$L12383~$L18476$L12343p~$L18475_gvar9 $SG12323_mppgcd $L12315}$L18474$L12286}$L18473ԉ$SG12172    _glcm0` _glcm  _gbezout  $SG12774_bezout   _content` $L12839w$SG12840 _lontyp$L12838$L12831e$L12825H$L12815$L12812$L18646  $L18709>$SG12934$L12921$L12898$L12897  $L12889M$L18712p$L18711, 7 $SG13043D$SG13009_revpol K V $SG13132$SG13088`k@ v  _gclone $SG132160 $L12645$L12648$L12639]$L18863T$L18865 $SG12961  $SG13344 $SG13339$SG13289  _polun$SG13468$SG13466$SG13420    $SG136114_Lazard _Lazard2`  & $SG13568T6 C` M ` j y $SG13719$SG13713t _det  $SG13788_fix_pol _srgcd $SG13914$SG13886  $SG13845p $SG12535 $L12555-$L12558 $L12550$L19122$L191244` $L19133\$L12594=$L12615 $L12605$L19136d$L19135t   $L13952L$SG13985$L13976_qf_disc $L13973$L13971o$L13970$L13955S$L19212d$L19211|_discsr  _smith  $SG140140 $SG14011 $SG14010 $SG14006*  $SG14119_gabs $SG14087h$SG14051`$SG14049X5 0 $SG14158$SG14150@  _quadgen _ginvmod  $SG14203J  $SG12751P $SG12748( $SG12738 _gvar $SG12729U  $SG12692\ d  _free _gdivgs p $SG14227 _polfnfp _cmp_pol $SG14357 z $SG14333    _ZX_disc  $SG14317   $SG14465  P _ratlift _to_frac  ` $SG14510 _nfgcdp _gdivise  _FpM _mulsi $SG14608   $SG14595  ! 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DmuE- }bv@|dN@H 'm!L(6 * i  0@P`0p@AHJLM\^`cd e0m@nPt`vpw} 0@P`p 0@P`p 0@P`p)8@Eeuclidean division (poldivres)euclidean division by zero (poldivres)normalizing a polynomial with 0 leading termprime too big in rootmod2factmodnot a prime in factmodfactmodfactormod for general polynomialsnot a prime in rootmodnot a prime in rootmodpolrootsmodfrobeniuskernelzero polynomial in FpXQ_pow. %Z not primespec_FpXQ_powvecpol_to_matpolpol_to_mat%Z not a prime in split_berlekampnew factorfactormodapprgenapprgenapprgen for p>=2^31incorrect coeffs in padic_pol_to_introotpadicrootpadicapprgen9apprgen9apprgen9 for p>=2^31factorpadicfactorpadicfactorpadic2 for non-monic polynomialfactorpadic2 (incorrect discriminant)factorpadicfactorpadicpolreversefactorpadicfactmod9polynomial variable must be of higher priority than finite field variable in factorfffactmod9[split9] time for splitting: %ld (%ld trials) need POLMOD coeffs in Kronecker_powmodneed Fq coeffs in Kronecker_powmodKronecker_powmodspec_Fq_pow_mod_polfactmod9factmod9factmod9rootsoldrootsoldtoo many iterations in rootsold(): using roots2() internal error in rootsold(): using roots2() roots2roots2too many iterations in rootstoo many iterations in roots2() ( laguer() ): real coefficients polynomial, using zrhqr() polynomial has probably multiple roots in zrhqr%ld polished roots = %Z* Finding eigenvalues too many iterations in hqr* Eigenvalues computed W|$u_á Vp+s j=t5~F@~^_F~^_Ð -, 4 = Cq=PD#@ܿ?ư>?.fileg..\..\..\..\OpenXM_contrib\pari-2.2\src\basemath\polarit1.c@comp.id# .drectve(pVU.text4 _setloop   _bot_avma.debug$FNe_incpos _incloop  _gdivise _gcmp0 _gmod ( 2 = M ] l _gadd _gcopy z _gmul _gun_ginv _gcmp1 _gneg_i  .data $SG9696H$SG9691 _zeropol $SG9680_gvar9 _gzero_gdiv    _free _top_divis _mulii    _polx$SG9983x.text0_stoi .debug$F_mod @ $SG9851 $SG9841$SG9837_bfffo $SG9833 $SG9962 _gdeux   _rootmod0  _sort _modii _addsi   _sqri _addii + : _FpX_gcd D $SG10063N _shifti X $SG10137b` u  _FpX_sub _FpX_red 0    $SG10190(_FpM_ker  $SG10188_zerocol _timer2 P" p"  .bss_modulo_deg1pol _gsmul$ _FpX_mul 0$ % _FpX  _cmpii _vals _gneg _split- _stopoly _gpowgs _try_pow/ _gcmp_1 _gsqr , 0 6 C`1 Vp2 $SG10499\e$SG104840o3 {3 3 5 $SG10676lP7  9 $SG10748|: $SG10904$SG10898_genrand    ? _divll _mulll pA _factmodpE G $SG11018_apprgenG + _grando0 $SG11185_gsub _egalii _ggval _poleval _ggcd $SG11159$SG111579L _gtrunc $SG11043_content K@M _gclone $SG11237 $SG11235VQ iR x _deriv @T @V  _lift V  Y _gtopoly $SG11511D$SG114758$SG11473,_getprec` ` ` $SG11713t$SG11703h$SG11701\e 0g  ph  _padicffi % _QM_inv 4 $SG11665A _dvmdii O _discsr Yl _caradj c _smith2 p { 0p _mulsi _Decomp _nilord   $SG11883$SG11877w y  $SG11831 z $SG11968pz _cmp_pol { | _gcmp P|  _gegal _powgi $SG12444X_gmodulo _polun ! $SG12417$SG12411+ _split9@ 6 $SG12185dBp $SG12083T d $SG12062$SG12049r _to_fq |  $SG12313 $L12362H$L12334W$SG12363 $L12358$SG12352$L12349$SG12348$L12345{$L18086t$L18085 $SG12817x $SG12755h$SG12754D_gmulsg _gconj _gaffect  _cmprr _gnorm _gshift _gop2z _gimag _gabs _greal _gexpo _gaffsg _divrs _mppi _gsigne $SG126328$SG12624,_roots2 $SG13085$SG13084$SG13082_gmul2n _gprec .rdata֟2@ _cgetc $SG12906$SG12902  __ftol _laguer _gexp _gopsg2 _gmax _gsqrt  _affrr _dbltor _zrhqr $SG13406t$SG13402l$SG13398<_gprec_w _balanc` ! _hqr $SG13339.rdata $Sv, .rdata T J $SG13291.rdata s=^;h .rdata  .rdata  $SG13269.data&_rtodbl _dowin32ctrlc_win32ctrlc_pari_err_poldivis_poldivres_gerepilemanysp_normalizepol_i_isinexactreal_gerepileupto_gmodulcp_isexactzero_gdivexact_rootmod2_hiremainder_mpinvmod_FpX_divres_gpmalloc_factmod_init_lift_intern0_root_mod_even_root_mod_2_root_mod_4_gerepile_mpsqrtmod_FpX_normalize_gmodulsg_ZX_s_add_FpXQ_pow_rootmod0_FpX_is_squarefree_derivpol_FpX_nbroots_FpX_is_totally_split_FpX_nbfact_Berlekamp_ker_msgtimer_DEBUGLEVEL_FpX_is_irred_FpV_roots_to_pol_divide_conquer_prod_trivfact_factcantor0_sort_factor_splitgen_stopoly_gen_init_pow_p_mod_pT_spec_FpXQ_pow_DEBUGMEM_factcantor_simplefactmod_mat_to_vecpol_vecpol_to_mat_mat_to_polpol_polpol_to_mat_split_berlekamp_small_to_pol_mymyrand_pol_to_small_split_berlekamp_addmul_factmod0_factormod0_gerepilecopy_padic_pol_to_int_rootpadic_hensel_lift_accel_rootpadiclift_FpX_eval_rootpadicliftroots_rootpadicfast_gunclone_padicsqrtnlift_powmodulo_apprgen9_cmp_padic_factorpadic2_padic_trivfact_pol_to_padic_int_to_padic_pvaluation_get_mul_table_get_bas_den_carrecomplet_allbase4_padicff2_element_mul_idealpows_primedec_factorpadic4_hensel_lift_fact_pol_to_monic_squarefree_polreverse_normalizepol_factorpadic0_FpX_rand_cmp_coeff_factmod9_forcecopy_Fp_factor_rel0_simplify_fqunclone_fprintferr_Kronecker_powmod_from_Kronecker_to_Kronecker_FqX_rand_init_pow_q_mod_pT_spec_Fq_pow_mod_pol_isabsolutepol_rootsold_flusherr_gops2gsz__real@8@40029a209a852f87e000_square_free_factorization__fltused_QuickNormL1_dummycopy__real@8@bffde000000000000000__real@8@3ffec000000000000000__real@8@3feb8637bd05af6c6800__real@8@3ffe8000000000000000__real@8@00000000000000000000?reel4@?1??gtodouble@@9@9ifactor1.obj/ 1002273803 100666 78563 ` L |;g.drectve(, .dataTe @0.texty Y P`.debug$FO*HB.bssA0.rdata@@@.rdata@@@-defaultlib:LIBCMT -defaultlib:OLDNAMES  I S_  !"#$%&'()*+,-./ %)+/5;=CGIOSYaegkmqy  @1r!!4  R&|.T8RDRddy>‚8[CfOd_ ?}j.T 'Xr0:l8G@ZV h ~dEXL PP0l ~@ 0!%h)-d2x7T=C8JRZc`myĆd((4MuQ08hI($t5|lZ>0? L^ ȟ T <\a .`e* .!e g;(g:i.`e*4e*e.;m*g*g&* / )+,+hk/0?u`?!*i?a??(???Miller-Rabin: testing base %ld miller(rabin)P.L.:factor O.K. Sorry false prime number %Z in plisprimesnextpr: prime %lu wasn't %lu mod 210 [caller of] snextprsnextpr: %lu should have been prime but isn't [caller of] snextprsnextpr: integer wraparound after prime %lu [caller of] snextprECM: number too small to justify this stage ECM: working on %ld curves at a time; initializing for one round for up to %ld rounds... ECM: stack tight, using clone space on the heap ECM: time = %6ld ms ECM: dsn = %2ld, B1 = %4lu, B2 = %6lu, gss = %4ld*420 ECM: time = %6ld ms, B1 phase done, p = %lu, setting up for B2 (got [2]Q...[10]Q) ECM: %lu should have been prime but isn't ellfacteur (got [p]Q, p = %lu = %lu mod 210) (got initial helix) ECM: time = %6ld ms, entering B2 phase, p = %lu ECM: finishing curves %ld...%ld (extracted precomputed helix / baby step entries) (baby step table complete) (giant step at p = %lu) ECM: time = %6ld ms, ellfacteur giving up. ECM: time = %6ld ms, p <= %6lu, found factor = %Z [caller of] elladd0Rho: searching small factor of %ld-bit integer Rho: searching small factor of %ld-word integer Rho: restarting for remaining rounds... Rho: using X^2%+1ld for up to %ld rounds of 32 iterations Rho: time = %6ld ms, Pollard-Brent giving up. Rho: time = %6ld ms, Pollard-Brent giving up. Rho: fast forward phase (%ld rounds of 64)... Rho: time = %6ld ms, %3ld rounds, back to normal mode found factor = %Z Rho: hang on a second, we got something here... Pollard-Brent failed. composite found %sfactor = %Z found factors = %Z, %Z, and %Z sRho: time = %6ld ms, %3ld round%s SQUFOF: entering main loop with forms (1, %ld, %ld) and (1, %ld, %ld) of discriminants %Z and %Z, respectively SQUFOF: blacklisting a = %ld on first cycle SQUFOF: blacklisting a = %ld on second cycle SQUFOF: first cycle exhausted after %ld iterations, dropping it SQUFOF: square form (%ld^2, %ld, %ld) on first cycle after %ld iterations, time = %ld ms SQUFOF: found factor 3 SQUFOF: found factor %ld^2 SQUFOF: squfof_ambig returned %ld SQUFOF: found factor %ld from ambiguous form after %ld steps on the ambiguous cycle, time = %ld ms SQUFOF: ...found nothing useful on the ambiguous cycle after %ld steps there, time = %ld ms SQUFOF: ...but the root form seems to be on the principal cycle SQUFOF: second cycle exhausted after %ld iterations, dropping it SQUFOF: square form (%ld^2, %ld, %ld) on second cycle after %ld iterations, time = %ld ms SQUFOF: found factor 5 SQUFOF: found factor %ld^2 SQUFOF: squfof_ambig returned %ld SQUFOF: found factor %ld from ambiguous form after %ld steps on the ambiguous cycle, time = %ld ms SQUFOF: ...found nothing useful on the ambiguous cycle after %ld steps there, time = %ld ms SQUFOF: ...but the root form seems to be on the principal cycle SQUFOF: giving up, time = %ld ms OddPwrs: is %Z ...a, or 3rd%s, or or 5th%s 7th power? modulo: resid. (remaining possibilities) 211: %3ld (3rd %ld, 5th %ld, 7th %ld) 209: %3ld (3rd %ld, 5th %ld, 7th %ld) 61: %3ld (3rd %ld, 5th %ld, 7th %ld) 203: %3ld (3rd %ld, 5th %ld, 7th %ld) 117: %3ld (3rd %ld, 5th %ld, 7th %ld) 31: %3ld (3rd %ld, 5th %ld, 7th %ld) 43: %3ld (3rd %ld, 5th %ld, 7th %ld) 71: %3ld (3rd %ld, 5th %ld, 7th %ld) But it nevertheless wasn't a cube. But it nevertheless wasn't a %ldth power. ifac_startfactoring 0 in ifac_startifac_reallocpartial impossibly short in ifac_reallocIFAC: new partial factorization structure (%ld slots) IFAC: main loop: repeated old factor %Z IFAC: unknown factor seen in main loopIFAC: main loop: repeated new factor %Z IFAC: main loop: another factor was divisible by %Z non-existent factor class in ifac_mainIFAC: after main loop: repeated old factor %Z sIFAC: main loop: %ld factor%s left IFAC: main loop: this was the last factor ifac_findpartial impossibly short in ifac_find`*where' out of bounds in ifac_findfactor has NULL exponent in ifac_findifac_sort_onepartial impossibly short in ifac_sort_one`*where' out of bounds in ifac_sort_one`washere' out of bounds in ifac_sort_onemisaligned partial detected in ifac_sort_oneIFAC: repeated factor %Z detected in ifac_sort_one composite equals prime in ifac_sort_oneprime equals composite in ifac_sort_oneifac_whoiswhopartial impossibly short in ifac_whoiswho`*where' out of bounds in ifac_whoiswhoavoiding nonexistent factors in ifac_whoiswhoIFAC: factor %Z is prime (no larger composite) IFAC: prime %Z appears with exponent = %ld compositeprimeIFAC: factor %Z is %s ifac_dividepartial impossibly short in ifac_divide`*where' out of bounds in ifac_dividedivision by composite or finished prime in ifac_dividedivision by nothing in ifac_divideIFAC: a factor was a power of another prime factor IFAC: a factor was divisible by another prime factor, leaving a cofactor = %Z IFAC: prime %Z appears at least to the power %ld IFAC: prime %Z appears with exponent = %ld ifac_crackpartial impossibly short in ifac_crack`*where' out of bounds in ifac_crackifac_crackoperand not known composite in ifac_crackIFAC: cracking composite %Z IFAC: checking for pure square IFAC: found %Z = %Z ^2 IFAC: factor %Z is prime IFAC: checking for odd power IFAC: found %Z = %Z ^%ld IFAC: factor %Z is prime IFAC: trying Pollard-Brent rho method first IFAC: trying Shanks' SQUFOF, will fail silently if input is too large for it. IFAC: trying Lenstra-Montgomery ECM IFAC: trying Multi-Polynomial Quadratic Sieve IFAC: forcing ECM, may take some time IFAC: unfactored composite declared prime %Z all available factoring methods failed in ifac_crackIFAC: factorizer returned strange object to ifac_crack factoringIFAC: factoring %Z yielded `factor' %Z which isn't! factoringIFAC: cofactor = %Z square not found by carrecomplet, ifac_crack recovering...IFAC: incorporating set of %ld factor(s)%s sorted them... stored (largest) factor no. %ld... ... factor no. %ld is a duplicate%s (so far)... factor no. %ld was unique%s ifac_decompfactoring 0 in ifac_decompIFAC: (Partial fact.)Stop requested. 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9  c c  c addhelpvSsinstallvrrD"",r,D"",s,killvSplotvV=GGIDGDGpplotboxvLGGplotclipvLplotcolorvLLplotcopyvLLGGD0,L,plotcursorLplotdrawvGD0,L,plotfilelsplothV=GGIpD0,L,D0,L,plothrawGGD0,L,plothsizesD0,L,plotinitvLD0,G,D0,G,D0,L,plotkillvLplotlinesvLGGD0,L,plotlinetypevLLplotmovevLGGplotpointsvLGGplotpointsizevLGplotpointtypevLLplotrboxvLGGplotrecthLV=GGIpD0,L,D0,L,plotrecthrawLGD0,L,plotrlinevLGGplotrmovevLGGplotrpointvLGGplotscalevLGGGGplotstringvLsD0,L,plottermlspsdrawvGD0,L,psplothV=GGIpD0,L,D0,L,psplothrawGGD0,L,typeGD"",r,addhelp(symbol,"message"): add/change help message for a symbolinstall(name,code,{gpname},{lib}): load from dynamic library 'lib' the function 'name'. Assign to it the name 'gpname' in this GP session, with argument code 'code'. If 'lib' is omitted use 'libpari.so'. If 'gpname' is omitted, use 'name'kill(x): kills the present value of the variable or function x. Returns new value or 0plot(X=a,b,expr,{ymin},{ymax}): crude plot of expression expr, X goes from a to b, with Y ranging from ymin to ymax. If ymin (resp. ymax) is not given, the minima (resp. the maxima) of the expression is used insteadplotbox(w,x2,y2): if the cursor is at position (x1,y1), draw a box with diagonal (x1,y1) and (x2,y2) in rectwindow w (cursor does not move)plotclip(w): clip the contents of the rectwindow to the bounding box (except strings)plotcolor(w,c): in rectwindow w, set default color to c. Possible values for c are 1=black, 2=blue, 3=sienna, 4=red, 5=cornsilk, 6=grey, 7=gainsboroughplotcopy(sourcew,destw,dx,dy,{flag=0}): copy the contents of rectwindow sourcew to rectwindow destw with offset (dx,dy). If flag's bit 1 is set, dx and dy express fractions of the size of the current output device, otherwise dx and dy are in pixels. dx and dy are relative positions of northwest corners if other bits of flag vanish, otherwise of: 2: southwest, 4: southeast, 6: northeast cornersplotcursor(w): current position of cursor in rectwindow wplotdraw(list, {flag=0}): draw vector of rectwindows list at indicated x,y positions; list is a vector w1,x1,y1,w2,x2,y2,etc. . If flag!=0, x1, y1 etc. express fractions of the size of the current output deviceplotfile(filename): set the output file for plotting output. "-" redirects to the same place as PARI outputploth(X=a,b,expr,{flags=0},{n=0}): plot of expression expr, X goes from a to b in high resolution. Both flags and n are optional. Binary digits of flags mean : 1 parametric plot, 2 recursive plot, 8 omit x-axis, 16 omit y-axis, 32 omit frame, 64 do not join points, 128 plot both lines and points, 256 use cubic splines, 512/1024 no x/y ticks, 2048 plot all ticks with the same length. n specifies number of reference points on the graph (0=use default value). Returns a vector for the bounding boxplothraw(listx,listy,{flag=0}): plot in high resolution points whose x (resp. y) coordinates are in listx (resp. listy). If flag is 1, join points, other non-0 flags should be combinations of bits 8,16,32,64,128,256 meaning the same as for ploth()plothsizes({flag=0}): returns array of 6 elements: terminal width and height, sizes for ticks in horizontal and vertical directions, width and height of characters. If flag=0, sizes of ticks and characters are in pixels, otherwise are fractions of the screen sizeplotinit(w,{x=0},{y=0},{flag=0}): initialize rectwindow w to size x,y. If flag!=0, x and y express fractions of the size of the current output device. x=0 or y=0 means use the full size of the deviceplotkill(w): erase the rectwindow wplotlines(w,listx,listy,{flag=0}): draws an open polygon in rectwindow w where listx and listy contain the x (resp. y) coordinates of the vertices. If listx and listy are both single values (i.e not vectors), draw the corresponding line (and move cursor). If (optional) flag is non-zero, close the polygonplotlinetype(w,type): change the type of following lines in rectwindow w. type -2 corresponds to frames, -1 to axes, larger values may correspond to something else. w=-1 changes highlevel plottingplotmove(w,x,y): move cursor to position x,y in rectwindow wplotpoints(w,listx,listy): draws in rectwindow w the points whose x (resp y) coordinates are in listx (resp listy). If listx and listy are both single values (i.e not vectors), draw the corresponding point (and move cursor)plotpointsize(w,size): change the "size" of following points in rectwindow w. w=-1 changes global valueplotpointtype(w,type): change the type of following points in rectwindow w. type -1 corresponds to a dot, larger values may correspond to something else. w=-1 changes highlevel plottingplotrbox(w,dx,dy): if the cursor is at (x1,y1), draw a box with diagonal (x1,y1)-(x1+dx,y1+dy) in rectwindow w (cursor does not move)plotrecth(w,X=xmin,xmax,expr,{flags=0},{n=0}): plot graph(s) for expr in rectwindow w, where expr is scalar for a single non-parametric plot, and a vector otherwise. If plotting is parametric, its length should be even and pairs of entries give points coordinates. If not, all entries but the first are y-coordinates. Both flags and n are optional. Binary digits of flags mean: 1 parametric plot, 2 recursive plot, 4 do not rescale w, 8 omit x-axis, 16 omit y-axis, 32 omit frame, 64 do not join points, 128 plot both lines and points. n specifies the number of reference points on the graph (0=use default value). Returns a vector for the bounding boxplotrecthraw(w,data,{flags=0}): plot graph(s) for data in rectwindow w, where data is a vector of vectors. If plot is parametric, length of data should be even, and pairs of entries give curves to plot. If not, first entry gives x-coordinate, and the other ones y-coordinates. Admits the same optional flags as plotrecth, save that recursive plot is meaninglessplotrline(w,dx,dy): if the cursor is at (x1,y1), draw a line from (x1,y1) to (x1+dx,y1+dy) (and move the cursor) in the rectwindow wplotrmove(w,dx,dy): move cursor to position (dx,dy) relative to the present position in the rectwindow wplotrpoint(w,dx,dy): draw a point (and move cursor) at position dx,dy relative to present position of the cursor in rectwindow wplotscale(w,x1,x2,y1,y2): scale the coordinates in rectwindow w so that x goes from x1 to x2 and y from y1 to y2 (y2=<;:9876543 2(100@/HP.,+*)('&%$#"!     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K Y _dvmdii k _addsi _gneg_i x  $SG10096T_hilb2nf  @ $SG10151 $SG10140 _poleval  $SG10119p@   _glog _gneg  $SG10306  _ZM_inv * _smith2 _hnfall 4 > K $SG10224U! $L10382"a _concat k _gnorm _denom _content x _isunit $L10377#"$SG10383$L10380"$L11653'$L11652'$SG103720' _gfloor  $SG10620_buchfu _gsubst  _isprime _factor _ggcd _gcmp_1 $SG105044 4 _buchall _polx_zpsoluble_pari_err_pvaluation_kronecker_dowin32ctrlc_win32ctrlc_hiremainder_qpsoluble_polrecip_qpsolublenf_gmodulcp_zidealstarinit_idealpows_idealval_checkprimeid_psquarenf_gmodulsg_idealnorm_basistoalg_psquare2nf_zideallog_lemma6nf_lemma7nf_psquare2qnf_concatsp_idealhermite_zpsolublenf_nfhilbertp_lift_intern0_element_powmodpr_nfmodprinit_element_div_element_pow_coefs_to_pol_nfhilbert_fprintferr_DEBUGLEVEL_idealfactor_nf_get_r1_nfhilbert0_bnfsunit_gerepilecopy_isprincipalfact_mathnfspec_factorback_i_dethnf_i_diagonal_isprincipal_checkbnf_bnfissunit_gerepile_element_val_algtobasis_rnfisnorm_inverseimage_primedec_vecconcat_bnfisnormbuch3.obj/ 1002273732 100666 89961 ` L {;9.drectve( .data-@@.text@!aw P`.debug$F04NHB.textP7C8 P`.debug$F88HB.text89 P`.debug$F_9o9HB.rdatay9@@@.rdata9@@@.rdata9@@@-defaultlib:LIBCMT -defaultlib:OLDNAMES vq -?̯? 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Testing primes <= B (= %ld) Testing primes | h(K) large Minkowski bound: certification will be VERY longMinkowski bound is too largePHASE 1: check primes to Zimmert bound = %ld **** Testing Different = %Z is %Z *** p = %ld Testing P = %Z #%ld in factor base is %Z Norm(P) > Zimmert bound End of PHASE 1. Default bound for regulator: 0.2 Default bound for regulator: 0.2 Mahler bound for regulator: %Z bnfcertifybug in lowerboundforregulatorM* = %Z pol = %Z old method: y = %Z, M0 = %Z M0 = %Z (lower bound for regulator) M = %Z Searching minimum of T2-form on units: BOUND = %ld .* [ %ld, %ld, %ld ]: %Z [ %ld, %ld, %ld ]: %Z [ %ld, %ld, %ld ]: %Z *** testing p = %ld p divides h(K) p divides w(K) Beta list = %Z generator of (Zk/Q)^*: %Z prime ideal Q: %Z column #%ld of the matrix log(b_j/Q): %Z new rank: %ld neither bnf nor bnr in conductor or discrayneither bnf nor bnr in conductor or discraybad subgroup in conductor or discrayincorrect subgroup in conductorrnfnormgroupnot an Abelian extension in rnfnormgroup?rnfnormgroup: bound for primes = %Z rnfnormgroup: prime bound too large, can't certifyrnfnormgroupnon Galois extension in rnfnormgrouprnfnormgroupnon Galois extension in rnfnormgroupnot an Abelian extension in rnfnormgrouprnfnormgrouprnfconductorincorrect subgroup in discrayincorrect character length in conductorofcharconductorofcharrayclassnolistdiscrayabslistfactordivexact is not exact!incorrect factorisation in decodemodulenon-positive bound in discrayabslistdiscrayabslistarchinternincorrect archimedean argument in discrayabslistarchinternr1>15 in discrayabslistarchinternStarting zidealstarunits computations Starting rayclassno computations %ld [1]: discrayabslistarchavma = %ld, t(z) = %ld avma = %ld, t(r) = %ld avma = %ld Starting rayclassno computations avma = %ld, t(r) = %ld avma = %ld zidealstarlistStarting discrayabs computations %ld [2]: discrayabslistarchavma = %ld, t(d) = %ld avma = %ld discrayabsdiscrayabslistlongsubgrouplistD$VPNQT$D$tNV t$ | hP^ÐD$&7D$0S+Í}D$@D$HT$D$,+Í +;v j=t5t j>T$@D$ρ$4;|&\$4T$$L$ Aʉ @;~L$D$,HL$D$,S=]+Ǎ+;v j=t5t j> $| D$$~D$0L$P4D$t%H;ʼnD$u%T$T$$>|L$DF+΋<8JuT$D$$L$4URT$ VPD$8QRP5D$4 ++;v j=t=t j>"|G͋IuD$PPL$D$Tq ++;v j=tt j>D$&3|\l$Ps+D$P .T$(jWQRPD$,PPL$@T$QJQPD$dHD$Pul$8D$+VPL$PVQPT$PPSRPPD$$jL$,PQVD$\D$8tID$PL$$|(|$PD$ PuG;~|$PL$VQ\$X5D$$+ƍ+;؉|$8v j=t=l$Ht j>D$ "l$(L$T$L$0xQWRUG } VD$$;|ZD$L$@t:PD$4 QUPVUVW u WVD$$C;~T$8D$L$P4@;D$Fl$H١ p+s j=tD$D5P"Ft j>D$  ˉT$ RWFF D$DVP_^][<Ð,L$0AQD$D$4T$t%HSD$AL$LU@VW-q%?D$0+ō+;|$\$v j=tt j>$3D$@l$u5U p+s j=t5~ F@nF؉FWVHuH|@Hl$;D$8;L$PT$TD$$-t$D$4|IL$(\$yT$@|$+T$@tPRPUƃ|$Kuʋ\$|$D$X|?|$,s+؋ 7T$0QRUPD$HD$LKuԋ\$|$D$DSPVL$, ;u D$4,D$TA@t$DD$TD$$L$ D$$(D$X;t:L$T$8A;ʉL$l$T$@BT$@l$&V_^][,ÐD$L$jPQ ÐhL$lSUVWQD$@V$hV\$(D$\t D$  %D$ v 3D$DD$,Ft$HHP L$@T$Pt%$jhWSl$ D$4@ H@ L$XD$`t%pWSt$@ .L$TD$0u$}+Í+;v j=t5t j>">t$,|6\$P~+ދD$0 ;T$PQRVtv7MuӋ$D$TD$Lp+Í +;v j=t=t j>L$";ȉ7|$DD$|AD$,\$P+؍p+D$P@D$3L$RPQ.D$\ HD$PuۋT$D$L;(t$l$`+ @Ћ+J(DuD$8D$Tu95x+s j=t=">5x+s j=tL$D="O T$HBT$@H$OWuD$T$HL$&S>B@PQ F|+T$\^MzD$(T$PQR MuD$XPPVPl$$H$u+Í+;L$`v j=t=t j>&7\$LT$@D$,+ڋ++ȍpT$\L$Xl$LT$\u)tPPQPL$(T$QPRL$d 1D$LHD$Lul$T$`D$8RPUPPWL$HPQPPT$jD$@RPWL$($D$LL$ttID$L$@|(t$D$4PuF;~t$D$Tl$MQRT$$WRt$$HL$X\$ D$ ȡ+ +|$\;v j=t=t j>D$"t$D$@ t$T;ljt$HL$h|$XD$T$C~ t$HT$TT$dt3++;v j=tN=xˍ+FNuCOt$d? 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Try to increase extra relationscleancolgetfu ***** IDEALS IN FACTORBASE ***** no %ld = %Z ***** IDEALS IN SUB FACTORBASE ***** ***** INITIAL PERMUTATION ***** vperm = %Z sub factorbase (%ld elements)Computing powers for sub-factor base: %ld powsubFBgen %ld ########## FACTORBASE ########## KC2=%ld, KC=%ld, KCZ=%ld, KCZ2=%ld, MAXRELSUP=%ld ++ idealbase[%ld] = %Zfactor base #### Looking for %ld relations (small norms) *** Ideal no %ld: %Z v[%ld]=%.0f BOUND = %.0f .*%4ldcglob = %ld. small_normfor this ideal small norm relationsElements of small norm gave %ld relations. 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V?????@ @6@aN@r@@,@,@@,A4AaAgA|AAAAAAA`B!B BB BtBBB"C*CuJC!tCC C CtCCCD)D5Dɵҵܵ &6>GNȶ޶-%y,Yx`ַwݷ=hT /7@Q_e&˸ݸ O&^dx>mt "X!9_v&%*GO[{  |˽}޽ }!3AKV`e~mt~˾,վ,߾2{dm*tKӿ&!  % 3a8lDF-%,pw=:S!] o z  ]zk FF FFF ! &~:MR~_ *MU^fxzF~ \u{&,McGl{F&.7=N,V\z,@GUIB!1/==N/G_^ ,qE cIVXx{VEOO%WoQ  t"6E,mx:79r#c, =c OnB Ek D ]\ZT  0@P"`*p+./AEGKLMNO Q0R@SPV`Wp`aly{ 0@P`p 0@P`p 0@P`p 0@ P ` p!"$-078 90>@CPD`EpFGIR\]`abfj n0t@uPz`}p~principalidealprincipalidealprincipalidealget_arch0 in get_arch_realidealdivprecision too low in ideal_better_basis (1)precision too low in ideal_better_basis (2)ideal_two_eltideal_two_eltideal_two_eltideal_two_elt, hard case: %ld zero ideal in idealfactoridealval entering idealaddtoone: x = %Z y = %Z ideals don't sum to Z_K in idealaddtooneideals don't sum to Z_K in idealaddtoone leaving idealaddtoone: %Z ideals don't sum to Z_K in idealaddtooneideals don't sum to Z_K in idealaddtooneincorrect idele in idealaddtoone entree dans element_invmodideal() : x = y = element_invmodideal sortie de element_invmodideal : v = entree dans idealaddmultoone() : list = not a list in idealaddmultooneideals don't sum to Z_K in idealaddmultooneideals don't sum to Z_K in idealaddmultoone sortie de idealaddmultoone v = generic conversion to finite fieldidealinvcannot invert zero idealcannot invert zero ideal0th power in idealpowprime_specnon-integral exponent in idealpownon-integral exponent in idealpowredcannot invert zero idealquotient not integral in idealdivexactideallllredideallllredentering idealllredLLL reductionnew ideal entree dans idealappr0() : x = not a prime ideal factorization in idealappr0 alpha = beta = alpha = sortie de idealappr0 p3 = entree dans idealchinese() : x = y = not a prime ideal factorization in idealchinesenot a suitable vector of elements in idealchinese sortie de idealchinese() : p3 = ideal_two_elt2element not in ideal in ideal_two_elt2element does not belong to ideal in ideal_two_elt2 entree dans idealcoprime() : x = y = sortie de idealcoprime() : p2 = element_reducenot a module in nfhermitenot a matrix in nfhermitenot a correct ideal list in nfhermitenot a matrix of maximal rank in nfhermitenot a matrix of maximal rank in nfhermitenot a matrix of maximal rank in nfhermitenfhermite, i = %ldnot a module in nfsmithnot a matrix in nfsmithnot a correct ideal list in nfsmithnot a matrix of maximal rank in nfsmithnot a matrix of maximal rank in nfsmithnfsmith for non square matricesbug2 in nfsmithnfsmithbug in nfsmithnfkermodprnfkermodpr, k = %ld / %ldnfsolvemodprnfsolvemodprnfsolvemodprincorrect dimension in nfsuplnot a module in nfdetintnot a matrix in nfdetintnot a correct ideal list in nfdetintnfdetintboth elements zero in nfbezoutnot a module in nfhermitemodnot a matrix in nfhermitemodnot a correct ideal list in nfhermitemod[1]: nfhermitemod[2]: nfhermitemod.fileg..\..\..\..\OpenXM_contrib\pari-2.2\src\basemath\base4.c@comp.id# .drectve(.text`5 D _gcmp0 _gcmp1 .debug$F{$ A ! / _avma90 M W f t _bot _gsigne  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"5:X^n[[3<U_\B'\,-?Ojx!+76NV_j7&5V`7|"7&'@J\`,kv%06JR[et[-+7S\c7 "+VY?I]enz<Bair~Y` 8 }Waqy*5CJTv|"(.5A-Jy%9AJP[Zr$>  8BV^gs|#LM~%+RZcox,Z,7(6?ENW_tf  8K}.7V^gmZ!'Z19YE$Pejr-T'PY z&0?ZPw,7GpPTS KTreating p^k = %Z^%ld allbaseallbasereducible polynomial in allbasedisc. factorisationROUND2: epsilon = %ld avma = %ld ordmaxrowred j=%ldTreating p^k = %Z^%ld Result for prime %Z is: %Z nfbasis00nfbasis00polynomial not in Z[X] in nfbasisnot a factorisation in nfbasis entering dedek with parameters p=%Z, f=%Z gcd has degree %ld entering Dedekind Basis with parameters p=%Z f = %Z, alpha = %Z new order: %Z entering Decomp with parameters: p=%Z, expo=%ld precision = %ld f=%Z leaving Decomp with parameters: f1 = %Z f2 = %Z e = %Z entering Nilord2 (factorization) entering Nilord2 (basis/discriminant) with parameters: p = %Z, expo = %ld fx = %Z, gx = %Z Fa = %ld and Ea = %ld beta = %Z gamma = %Z Increasing Fa bug in nilord (no root). Is p a prime ?bug in nilord (no suitable root), is p = %Z a prime? Increasing Ea newtoncharpolynewtonsumsnewtonsums: result too large to fit in cache non separable polynomial in update_alpha! factmodsimple primedecunramified factorspradicalprimedec (bad pradical)h[%ld]kerlenskerlens2prime_two_elt_loop, hard case: %d mymod (missing type)incorrect polynomial in rnf functionrnf functionnon-monic relative polynomialsincorrect variable in rnf functionincorrect polcoeff in rnf functionIdeals to consider: %Z^%ld treating %Z %ld%s pass new order: rnfordmaxnot a pseudo-basis in nfsimplifybasisnot a pseudo-matrix in rnfdetnot a pseudo-matrix in rnfsteinitzrnfsteinitznot a pseudo-matrix in rnfbasisnot a pseudo-matrix in rnfbasisnot a pseudo-matrix in rnfisfreepolcompositum0compositumnot the same variable in compositumcompositumcompositumcompositum: %Z not separablecompositum: %Z not separablernfequationrnfequationnot k separable relative equation in rnfequationnot a pseudo-matrix in rnflllgramrnflllgramkk = %ld %ld %ld rnflllgramrnfpolredrelative basis computed 8n.lJ fg :q 3[e!5 1 A>/  Ms(R$v>GL/e ^cYz,~   " ,02@4P6`8pEGHIJMNOPQV W0X@aPf`gphijklmnu|~ 0@P`p 0@P`p  0@P`p!$%26FGHJ K0P@SPU`VpZ\^aopqu|}~ 0@P`p.fileg..\..\..\..\OpenXM_contrib\pari-2.2\src\basemath\base2.c@comp.id# .drectve(&9!.text 46_allbase _gdiv _dvmdii _sqri  _gzero_egalii _gtrans _mulii _gmul  .data- $SG10376 _gun_idmat , :F _bot_avma.debug$Fl,0'P@ d $SG9674Hn z $SG9669(_ZX_disc _timer2 $SG9659 $SG9655_rquot   _addsi _addii _cmpii _shifti   _gneg _mulmatiP   _rtran _gadd _ordmaxP  $SG10324_ggval _mppgcd _gcmp1 _divis _modii $SG10130\_cmpsi _bfffo _rrmdr`  _powmati@   _rowred  $SG10001_mtran! 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Change of variables discardedplease apply nfinit firstincorrect bigray fieldplease apply bnrinit(,,1) and not bnrinit(,)missing units in %sincorrect matrix for idealincorrect bigidealincorrect prime idealincorrect prhall formatpolynomial not in Z[X] in %sincompatible modulus in %s: mod = %Z, nf = %ZtschirnhaustschirnhausTschirnhaus transform. New pol: %Zgpolcomp (different degrees)galoisgaloisgalois of degree higher than 11galoisgalois of reducible polynomialgalois (bug1)galois (bug4)galois (bug3)galois (bug2)incorrect galois automorphism in galoisapplygaloisapplygaloisapplygaloisapplygaloisapplynfiso or nfinclmatrix Mmatrix MCfalse nf in nf_get_r1false nf in nf_get_r2mult. tablematricesinitalgall0initalgnfinitnon-monic polynomial. 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