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Revision 1.4, Tue Oct 1 06:59:14 2002 UTC (21 years, 9 months ago) by noro
Branch: MAIN
Changes since 1.3: +10839 -10762 lines 

Sorry, libpari.lib, libpariw_lib were broken.
We use the old version of these files.

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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! Using timer2()timer %ld wasn't in useTime : %ld
Time : %ld
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U@0-defaultlib:LIBCMT -defaultlib:OLDNAMES Euler=Euler(): Euler's constant with current precisionI=I(): square root of -1List({x=[]}): transforms the vector or list x into a list. Empty list if x is omittedMat({x=[]}): transforms any GEN x into a matrix. Empty matrix if x is omittedMod(x,y,{flag=0}): creates the object x modulo y. flag is optional, and can be 0: default, creates on the Pari stack, or 1: creates a permanent object on the heapO(a^b): p-adic or power series zero with precision given by bPi=Pi(): the constant pi, with current precisionPol(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the leading coefficientPolrev(x,{v=x}): convert x (usually a vector or a power series) into a polynomial with variable v, starting with the constant termQfb(a,b,c,{D=0.}): binary quadratic form a*x^2+b*x*y+c*y^2. D is optional (0.0 by default) and initializes Shanks's distance if b^2-4*a*c>0Ser(x,{v=x}): 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 coefficients. 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 -pi<arg(x)<=piasin(x): inverse sine of xasinh(x): inverse hyperbolic sine of xatan(x): inverse tangent of xatanh(x): inverse hyperbolic tangent of xbernfrac(x): Bernoulli number B_x, as a rational numberbernreal(x): Bernoulli number B_x, as a real number with the current precisionbernvec(x): Vector of rational Bernoulli numbers B_0, B_2,...up to B_(2x)besseljh(n,x): J-bessel function of index n+1/2 and argument x, where n is a non-negative integerbesselk(nu,x,{flag=0}): K-bessel function of index nu and argument x (x positive real of type real, nu of any scalar type). flag is optional, and may be set to 0: default; 1: use hyperubestappr(x,k): gives the best approximation to the real x with denominator less or equal to kbezout(x,y):  gives a 3-dimensional row vector [u,v,d] such that d=gcd(x,y) and u*x+v*y=dbezoutres(x,y):  gives a 3-dimensional row vector [u,v,d] such that d=resultant(x,y) and u*x+v*y=d, where x and y are polynomialsbigomega(x): number of prime divisors of x, counted with multiplicitybinary(x): gives the vector formed by the binary digits of x (x integer)binomial(x,y): binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any xbitand(x,y): bitwise "and" of two integers x and y.  Negative numbers behave as if modulo big power of 2bitneg(x,{n=-1}): bitwise negation of an integers x truncated to n bits.  n=-1 means represent infinite sequences of bit 1 as negative numbers.  Negative numbers behave as if modulo big power of 2bitnegimpy(x,y): bitwise "negated imply" of two integers x and y, in other words, x BITAND BITNEG(y).  Negative numbers behave as if modulo big power of 2bitor(x,y): bitwise "or" of two integers x and y.  Negative numbers behave as if modulo big power of 2bittest(x,n): gives bit number n (coefficient of 2^n) of the integer xbitxor(x,y): bitwise "exclusive or" of two integers x and y.  Negative numbers behave as if modulo big power of 2bnfcertify(bnf): certify the correctness (i.e. remove the GRH) of the bnf data output by bnfclassunit or bnfinitbnfclassunit(P,{flag=0},{tech=[]}): compute the class group, regulator of the number field defined by the polynomial P, and also the fundamental units if they are not too large. flag and tech are both optional. flag can be any of 0: default, 1: insist on having fundamental units, 2: do not compute units. See manual for details about tech. P may also be a non-zero integer, and is then considered as the discriminant of a quadratic orderbnfclgp(P,{tech=[]}): compute the class group 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 techbnfdecodemodule(nf,fa): given a coded module fa as in bnrdisclist, gives the true modulebnfinit(P,{flag=0},{tech=[]}): compute the necessary data for future use in ideal and unit group computations, including fundamental units if they are not too large. flag and tech are both optional. flag can be any of 0: default, 1: insist on having fundamental units, 2: do not compute units, 3: small bnfinit, which can be converted to a big one using bnfmake. See manual for details about techbnfisintnorm(bnf,x): compute a complete system of solutions (modulo units of positive norm) of the absolute norm equation N(a)=x, where a belongs to the maximal order of big number field bnf (if bnf is not certified, this depends on GRH)bnfisnorm(bnf,x,{flag=1}): Tries to tell whether x (in Q) is the norm of some fractional y (in bnf). Returns a vector [a,b] where x=Norm(a)*b. Looks for a solution which is a S-unit, with S a certain list of primes (in bnf) containing (among others) all primes dividing x. If bnf is known to be Galois, set flag=0 (in this case, x is a norm iff b=1). If flag is non zero the program adds to S all the primes : dividing flag if flag<0, or 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(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<y)lift(x,{v}): lifts every element of Z/nZ to Z or T[x]/PT[x] to T[x] for a type T if v is omitted, otherwise lift only polymods with main variable v. If v does not occur in x, lift only intmodslindep(x,{flag=0}): Z-linear dependencies between components of x. flag is optional, and can be 0: default, using Hastad et al; positive, and in that case should be between 0.5 and 1.0 times the accuracy in decimal digits of x, using a standard LLL; or negative in which case it returns, once a relation has been foundlistcreate(n): creates an empty list of length nlistinsert(list,x,n): insert x at index n in list, shifting the remaining elements to the rightlistkill(list): kills listlistput(list,x,{n}): sets n-th element of list equal to x. If n is omitted or greater than the current list length, just append xlistsort(list,{flag=0}): sort list in place. If flag is non-zero, suppress all but one occurence of each element in listlngamma(x): logarithm of the gamma function of xlog(x,{flag=0}): natural logarithm of x. flag is optional, and can be 0: default, or 1: calls logagm, computed using the agmmatadjoint(x): adjoint matrix of xmatalgtobasis(nf,x): nfalgtobasis applied to every element of the matrix xmatbasistoalg(nf,x): nfbasistoalg applied to every element of the matrix xmatcompanion(x): companion matrix to polynomial xmatdet(x,{flag=0}): determinant of the matrix x using Gauss-Bareiss. If (optional) flag is set to 1, use classical gaussian elimination (slightly better for integer entries)matdetint(x): some multiple of the determinant of the lattice generated by the columns of x (0 if not of maximal rank). Useful with mathnfmodmatdiagonal(x): creates the diagonal matrix whose diagonal entries are the entries of the vector xmateigen(x): eigenvectors of the matrix x given as columns of a matrixmathess(x): Hessenberg form of xmathilbert(n): Hilbert matrix of order n (n C-integer)mathnf(x,{flag=0}): (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x. flag is optional whose value range from 0 to 4 (0 if omitted), meaning : 0: naive algorithm. 1: Use Batut's algorithm. Output 2-component vector [H,U] such that H is the HNF of x, and U is a unimodular matrix such that xU=H. 2: Use Havas's algorithm. Output 3-component vector [H,U,P] such that H is the HNF of x, U is unimodular and P is a permutation of the rows such that P applied to xU gives H. 3: Use Batut's algorithm. Output [H,U,P] as in 2mathnfmod(x,d): (upper triangular) Hermite normal form of x, basis for the lattice formed by the columns of x, where d is a multiple of the non-zero determinant of this latticemathnfmodid(x,d): (upper triangular) Hermite normal form of x concatenated with d times the identity matrixmatid(n): identity matrix of order n (n C-integer)matimage(x,{flag=0}): basis of the image of the matrix x. flag is optional and can be set to 0 or 1, corresponding to two different algorithmsmatimagecompl(x): vector of column indices not corresponding to the indices given by the function matimagematindexrank(x): gives two extraction vectors (rows and columns) for the matrix x such that the extracted matrix is square of maximal rankmatintersect(x,y): intersection of the vector spaces whose bases are the columns of x and ymatinverseimage(x,y): an element of the inverse image of the vector y by the matrix x if one exists, the empty vector otherwisematisdiagonal(x): true(1) if x is a diagonal matrix, false(0) otherwisematker(x,{flag=0}): basis of the kernel of the matrix x. flag is optional, and may be set to 0: default; non-zero: x is known to have integral entriesmatkerint(x,{flag=0}): LLL-reduced Z-basis of the kernel of the matrix x with integral entries. flag is optional, and may be set to 0: default, uses a modified LLL, 1: uses matrixqz3 and the HNF, 2: uses another modified LLLmatmuldiagonal(x,d): product of matrix x by diagonal matrix whose diagonal coefficients are those of the vector d, equivalent but faster than x*matdiagonal(d)matmultodiagonal(x,y): product of matrices x and y, knowing that the result will be a diagonal matrix. Much faster than general multiplication in that casematpascal(n,{q}): Pascal triangle of order n if q is omited. q-Pascal triangle otherwisematrank(x): rank of the matrix xmatrix(m,n,{X},{Y},{expr=0}): mXn matrix of expression expr, the row variable X going from 1 to m and the column variable Y going from 1 to n. By default, fill with 0smatrixqz(x,p): if p>=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$J(I,H0G4F8E<D@CDBHAL@P?T>X=\<`;d:h9l8p7t6x5|43210/.-,+*)('&%$#"! 
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0						R		QOabsGpacosGpacoshGpaddellGGGaddprimesGadjGagmGGpakellGGalgdepGLpalgdep2GLLpalgtobasisGGanellGLapellGGapell2GGapprpadicGGargGpasinGpasinhGpassmatGatanGpatanhGpbasisGfbasis2GfbasistoalgGGbernrealLpbernvecLbestapprGGpbezoutGGbezoutresGGbigomegaGbilhellGGGpbinGLbinaryGbittestGGboundcfGLboundfactGLbuchcertifylGbuchfuGpbuchgenGD0.3,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 -pi<arg(x)<=piasin(x)=inverse sine of xasinh(x)=inverse hyperbolic sine of xassmat(x)=associated matrix to polynomial xatan(x)=inverse tangent of xatanh(x)=inverse hyperbolic tangent of xbasis(x)=integral basis of the field Q[a], where a is a root of the polynomial x, using the round 4 algorithmbasis2(x)=integral basis of the field Q[a], where a is a root of the polynomial x, using the round 2 algorithmbasistoalg(nf,x)=transforms the vertical vector x on the integral basis into an algebraic numberbernreal(x)=Bernoulli number B_x, as a real number with the current precisionbernvec(x)=Vector of rational Bernoulli numbers B_0, B_2,... up to B_(2x)bestappr(x,k)=gives the best approximation to the real x with denominator less or equal to kbezout(x,y)= gives a 3-dimensional row vector [u,v,d] such that d=gcd(x,y) and u*x+v*y=dbezoutres(x,y)= gives a 3-dimensional row vector [u,v,d] such that d=resultant(x,y) and u*x+v*y=d, where x and y are polynomialsbigomega(x)=number of repeated prime divisors of xbilhell(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 pointsbin(x,y)=binomial coefficient x*(x-1)...*(x-y+1)/y! defined for y in Z and any xbinary(x)=gives the vector formed by the binary digits of x (x C-integer)bittest(x,n)=gives bit number n (coefficient of 2^n) of the integer xboundcf(x,lmax)=continued fraction expansion of x with at most lmax termsboundfact(x,lim)=partial factorization of the integer x (using primes up to lim)buchcertify(bnf)=certify the correctness (i.e. remove the GRH) of the bnf data output by buchinit or buchinitfubuchfu(bnf)=compute the fundamental units of the number field bnf output by buchinitbuchgen(P,...)=compute the structure of the class group and the regulator for the number field defined by the polynomial P. See manual for the other parameters (which can be omitted)buchgenforcefu(P,...)=compute the structure of the class group, the regulator a primitive root of unity and a system of fundamental units for the number field defined by the polynomial P, and insist until the units are obtained. See manual for the other parameters (which can be omitted)buchgenfu(P,...)=compute the structure of the class group, the regulator a primitive root of unity and a system of fundamental units (if they are not too large) for the number field defined by the polynomial P. See manual for the other parameters (which can be omitted)buchimag(D,...)=compute the structure of the class group of the complex quadratic field of discriminant D<0. See manual for the other parameters (which can be omitted)buchinit(P,...)=compute the necessary data for future use in ideal and unit group computations. See manual for detailsbuchinitforcefu(P,...)=compute the necessary data for future use in ideal and unit group computations, and insist on having fundamental units. See manual for detailsbuchinitfu(P,...)=compute the necessary data for future use in ideal and unit group computations, including fundamental units if they are not too large. See manual for detailsbuchnarrow(bnf)=given a big number field as output by buchinitxx, gives as a 3-component vector the structure of the narrow class groupbuchray(bnf,ideal)=given a big number field as output by buchinitfu (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 modulebuchrayinit(bnf,ideal)=same as buchrayinitgen, except that the generators are not explicitly computedbuchrayinitgen(bnf,ideal)=given a big number field as output by buchinitfu (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 for computing in the ray class group  corresponding to this module. In particular, the fifth component is the ray class group structurebuchreal(D,...)=compute the structure of the class group and the regulator of the real quadratic field of discriminant D>0 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<y)lexsort(x)=sort the elements of the vector x in ascending lexicographic orderlift(x)=lifts every element of Z/nZ to Z or Z[x]/PZ[x] to Z[x]lindep(x)=Z-linear dependencies between components of x (Hastad et al)lindep2(x,dec)=Z-linear dependencies between components of x using LLL, where dec should be about one half the number of decimal digits of precisionlll(x)=lll reduction of the vectors forming the matrix x (gives the unimodular transformation matrix)lll1(x)=old version of lll reduction of the vectors forming the matrix x (gives the unimodular transformation matrix)lllgen(x)=lll reduction of the vectors forming the matrix x with polynomial coefficients (gives the unimodular transformation matrix)lllgram(x)=lll reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix)lllgram1(x)=old version of lll reduction of the lattice whose gram matrix is x (gives the unimodular transformation matrix)lllgramgen(x)=lll reduction of the lattice whose gram matrix is x with polynomial coefficients (gives the unimodular transformation matrix)lllgramint(x)=lll reduction of the lattice whose gram matrix is the integral matrix x (gives the unimodular transformation matrix)lllgramkerim(x)=kernel and lll reduction of the lattice whose gram matrix is the integral matrix xlllgramkerimgen(x)=kernel and lll reduction of the lattice whose gram matrix is the matrix x with polynomial coefficientslllint(x)=lll reduction of the vectors forming the matrix x when the gram matrix is integral (gives the unimodular transformation matrix)lllintpartial(x)=partial (hence faster) lll reduction of the vectors forming the matrix x when the gram matrix is integral (gives the unimodular transformation matrix)lllkerim(x)=kernel and lll reduction of the vectors forming the integral matrix xlllkerimgen(x)=kernel and lll reduction of the vectors forming the matrix x with polynomial coefficientslllrat(x)=lll reduction of the vectors forming the matrix x, computations done with rational numbers (gives the unimodular transformation matrix)ln(x)=log(x)=natural logarithm of xlngamma(x)=logarithm of the gamma function of xlocalred(e,p)= e being an ellliptic 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_plog(x)=ln(x)=natural logarithm of xlogagm(x)=natural logarithm of x, computed using agm (faster than log for more than a few hundred decimal digits)lseriesell(e,s,N,A)=L-series at s of the elliptic curve e, where |N| is the conductor, sign(N) the sign of the functional equation, and A a cut-off point close to 1makebigbnf(sbnf)=transforms small sbnf as output by smallbuchinit into a true big bnfmat(x)=transforms any GEN x into a matrixmatextract(x,y,z)=extraction of the components of the matrix x according to the vector or masks y (for the rows) and z (for the columns) from left to right (1,2,4,8,...for the first, second, third, fourth, ...rows or columns)mathell(e,x)=gives the height matrix for vector of points x on elliptic curve e using theta functionsmatrix(m,n,X,Y,expr)=mXn matrix of expression expr, the row variable X going  from 1 to m and the column variable Y going from 1 to nmatrixqz(x,p)=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 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													



 
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TTRQ@@@@@?thueinitinvalid polynomial in thue (need deg>2)thueinitinvalid polynomial in thue (need deg>2)non-monic polynomial in thuec1 = %Z
c2 = %Z
thueepsilon_3 -> %Z
incorrect system of unitsnot a tnf in thuex1 -> %Z
Entering CF
Increasing precision
Semirat. reduction ok. 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 !!
B0 -> %Z
thue (totally rational case)Check_smallexpected an integer in bnfisintnorm%Z eliminated because of sign
gcd f_P  does not divide n_p
sol = %Z
Partial = %Z
looking for a fundamental unit of norm -1
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Rg>hbRgYil> '0(@8P<`>p@UVWZ\^_acd p0s@uP{`}pnffactormodpolynomial variable must have highest priority in nffactormodcannot find a suitable factor in localpolnffactormodpolynomial variable must have highest priority in nffactormodnffactormod%Z not a prime (nffactormod)division by zero in nfmod_pol_divresnot a polynomial in nfrootspolynomial variable must have highest priority in nfrootstest if the polynomial is square-free
nffactorpolynomial variable must have highest priority in nffactortest if the polynomial is square-free
number of factor(s) found: %ld
square-freenfsqffthe bound on the T2-norm of the coeff. is: %Z
lower bound for the prime 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$VС++;v
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A@"@2@*@@@333333@A@"@2@*@@@@`6A@@"@2@.@@@@`6A@"@2@.@@@@`6A@"@3@0@@@	@`6A@"@3@0@@@	@`6A@"@3@0@@@	@`6Av@"@3@0@@@	@`6Aj@"@4@1@@@ffffff
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@`FA@$@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
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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$uuQPQPËL$uuPQQPÐ4\?[c[nZvY.fileg..\..\..\..\OpenXM_contrib\pari-2.2\src\basemath\galconj.c@comp.id#
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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:
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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)
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   %3ld factor of degree %3ld
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 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
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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.
[2] ifac_decompsIFAC: found %ld large prime (power) factor%s.
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highlvl.obj/    1002273799              100666  18309     `
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caddhelpvSsinstallvrrD"",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<y1 is allowed)plotstring(w,x,{flags=0}): draw in rectwindow w the string corresponding to x.  Bits 1 and 2 of flag regulate horizontal alignment: left if 0, right if 2, center if 1.  Bits 4 and 8 regulate vertical alignment: bottom if 0, top if 8, v-center if 4. Can insert additional gap between point and string: horizontal if bit 16 is set, vertical if bit 32 is setplotterm("termname"): set terminal to plot in high resolution to. Ignored by some drivers. In gnuplot driver possible terminals are the same as in gnuplot, terminal options can be put after the terminal name and space; terminal size can be put immediately after the name, as in "gif=300,200". If term is "?", lists possible values. Positive return value means successpsdraw(list, {flag=0}): same as plotdraw, except that the output is a postscript program in psfile (pari.ps by default), and flag!=0 scales the plot from size of the current output device to the standard postscript plotting sizepsploth(X=a,b,expr,{flags=0},{n=0}): same as ploth, except that the output is a postscript program in psfile (pari.ps by default)psplothraw(listx,listy,{flag=0}): same as plothraw, except that the output is a postscript program in psfile (pari.ps by default)type(x,{t}): if t is not present, output the type of the GEN x. Else make a copy of x with type t. Use with extreme care, usually with t = t_FRACN or t = t_RFRACN). Try \t for a list of typeslibpari.dllcouldn't open dynamic library '%s'couldn't open dynamic symbol table of processcan't find symbol '%s' in library '%s'can't find symbol '%s' in dynamic symbol table of process[secure mode]: about to install '%s'. OK ? (^C if not)
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gen3.obj/       1002273798              100666  94693     `
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PHASE 2: are all primes good ?

  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
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#### Computing fundamental units
getfunot enough relations for fundamental unitsfundamental units too largeinsufficient precision for fundamental unitsunknown problem with fundamental units%s, not givenred_mod_unitsnot a vector/matrix in cleancolnot the same number field in isprincipalzero ideal in isprincipalzero ideal in isprincipalisprincipalall0precision too low for generators, e = %ldprecision too low for generators, not givennbtest = %ld, ideal = %Z
split_ideal: increasing factor base [%ld]
isprincipal (incompatible bnf generators)insufficient precision for generators, not givenisprincipalall0not the same number field in isunitnot an algebraic number in isunitisunitisunitadding vector = %Z
vector in new basis = %Z
list = %Z
base change matrix =
completing bnf (building cycgen)completing bnf (building matal)*%ld %ld makematal
bug in codeprime/smallbuchinitbnfnewpreclog_poleval
#### Computing class group generators
classgroup generatorsincorrect sbnf in bnfmakebnfclassunitincorrect parameters in classgroupclassgroupallbnfinitnot enough relations in bnfxxxinitalg & rootsof1N = %ld, R1 = %ld, R2 = %ld, RU = %ld
D = %Z
buchall (%s)LLLnfLIMC = %ld, LIMC2 = %ld
relsup = %ld, ss = %ld, KCZ = %ld, KC = %ld, KCCO = %ld
After trivial relations, cglob = %ld
small_norm
#### Looking for random relations
*** Increasing sub factor base
s
(need %ld more relation%s)
random_relationbuchallregulator is zero.
bestappr
#### Tentative class number: %Z
compute_check
 ***** check = %f
suspicious check. 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.
Computing rank: %ld; independent columns: %Z

nb. fact./nb. small norm = %ld/%ld = %.3f

nb. small norm = 0
prec too low in red_ideal[%ld]: %ldBound for norms = %.0f

rel = %ld^%ld 

Rank = %ld
relations = 

matarch = %Z
before hnfadd:
vectbase[vperm[]] = 
[,]~
looking hard for %Z
phase=%ld,jideal=%ld,jdir=%ld,rand=%ld
.Upon exit: jideal=%ld,jdir=%ld

idealpro = %Z

++++ cglob = %ld: new relation (need %ld)(jideal=%ld,jdir=%ld,phase=%ld)for this relationarchimedian part = %Z
rel. cancelled. phase %ld: %ld (jideal=%ld,jdir=%ld)Be honest for primes from %ld to %ld
%ld 
be honest
#### Computing regulator

#### Computing check
c = %Z
den = %Z
bestappr/regulatorweighted T2 matricesincorrect matrix in relationranknot a maximum rank matrix in relationrankrelationrank{G'@?(@+m0_?&DT!@SVW|$uA\$t%@+t
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chk_gen_init: subfield %Z
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chk_gen_init: estimated prec = %ld (initially %ld)
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ordmaxrowred j=%ldTreating p^k = %Z^%ld
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%Z
nfbasis00nfbasis00polynomial not in Z[X] in nfbasisnot a factorisation in nfbasis  entering dedek with parameters p=%Z,
  f=%Z
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  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
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 new order:
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nfinit_reduceyou have found a counter-example to a conjecture, please send us
the polynomial as soon as possiblepolmax = %Z
nfinitincorrect nf in nfnewprecrootsof1rootsof1rootsof1 (bug1)rootsof1not an integer type in dirzetaktoo many terms in dirzetak %ld
discriminant too large for initzeta, sorry
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