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Import PARI/GP 2.0.17 beta.

/*******************************************************************/
/*                                                                 */
/*               S-CLASS GROUP AND NORM SYMBOLS                    */
/*          (Denis Simon, desimon@math.u-bordeaux.fr)              */
/*                                                                 */
/*******************************************************************/
/* $Id: buch4.c,v 1.1.1.1 1999/09/16 13:47:29 karim Exp $ */
#include "pari.h"

static long
psquare(GEN a,GEN p)
{
  long v;
  GEN ap;

  if (gcmp0(a) || gcmp1(a)) return 1;

  if (!cmpis(p,2))
  {
    v=vali(a); if (v&1) return 0;
    return (smodis(shifti(a,-v),8)==1);
  }

  ap=stoi(1); v=pvaluation(a,p,&ap);
  if (v&1) return 0;
  return (kronecker(ap,p)==1);
}

static long
lemma6(GEN pol,GEN p,long nu,GEN x)
{
  long i,lambda,mu,ltop=avma;
  GEN gx,gpx;

  for (i=lgef(pol)-2,gx=(GEN) pol[i+1]; i>1; i--)
    gx=addii(mulii(gx,x),(GEN) pol[i]);
  if (psquare(gx,p)) return 1;

  for (i=lgef(pol)-2,gpx=mulis((GEN) pol[i+1],i-1); i>2; i--)
    gpx=addii(mulii(gpx,x),mulis((GEN) pol[i],i-2));

  lambda=pvaluation(gx,p,&gx);
  if (gcmp0(gpx)) mu=BIGINT; else mu=pvaluation(gpx,p,&gpx);
  avma=ltop;

  if (lambda>(mu<<1)) return 1;
  if (lambda>=(nu<<1) && mu>=nu) return 0;
  return -1;
}

static long
lemma7(GEN pol,long nu,GEN x)
{ long i,odd4,lambda,mu,mnl,ltop=avma;
  GEN gx,gpx,oddgx;

  for (i=lgef(pol)-2,gx=(GEN) pol[i+1]; i>1; i--)
    gx=addii(mulii(gx,x),(GEN) pol[i]);
  if (psquare(gx,gdeux)) return 1;

  for (i=lgef(pol)-2,gpx=gmulgs((GEN) pol[i+1],i-1); i>2; i--)
    gpx=gadd(gmul(gpx,x),gmulgs((GEN) pol[i],i-2));

  lambda=vali(gx);
  if (gcmp0(gpx)) mu=BIGINT; else mu=vali(gpx);
  oddgx=shifti(gx,-lambda);
  mnl=mu+nu-lambda;
  odd4=smodis(oddgx,4);
  avma=ltop;
  if (lambda>(mu<<1)) return 1;
  if (nu > mu)
    { if (mnl==1 && (lambda&1) == 0) return 1;
      if (mnl==2 && (lambda&1) == 0 && odd4==1) return 1;
    }
  else
    { if (lambda>=(nu<<1)) return 0;
      if (lambda==((nu-1)<<1) && odd4==1) return 0;
    }
  return -1;
}

static long
zpsol(GEN pol,GEN p,long nu,GEN pnu,GEN x0)
{
  long i,result,ltop=avma;
  GEN x,pnup;

  result = (cmpis(p,2)) ? lemma6(pol,p,nu,x0) : lemma7(pol,nu,x0);
  if (result==+1) return 1; if (result==-1) return 0;
  x=gcopy(x0); pnup=mulii(pnu,p);
  for (i=0; i<itos(p); i++)
  {
    x=addii(x,pnu);
    if (zpsol(pol,p,nu+1,pnup,x)) { avma=ltop; return 1; }
  }
  avma=ltop; return 0;
}

/* vaut 1 si l'equation y^2=Pol(x) a une solution p-adique entiere
 * 0 sinon. Les coefficients sont entiers.
 */
long
zpsoluble(GEN pol,GEN p)
{
  if ((typ(pol)!=t_POL && typ(pol)!=t_INT) || typ(p)!=t_INT )
    err(typeer,"zpsoluble");
  return zpsol(pol,p,0,gun,gzero);
}

/* vaut 1 si l'equation y^2=Pol(x) a une solution p-adique rationnelle
 * (eventuellement infinie), 0 sinon. Les coefficients sont entiers.
 */
long
qpsoluble(GEN pol,GEN p)
{
  if ((typ(pol)!=t_POL && typ(pol)!=t_INT) || typ(p)!=t_INT )
    err(typeer,"qpsoluble");
  if (zpsol(pol,p,0,gun,gzero)) return 1;
  return (zpsol(polrecip(pol),p,1,p,gzero));
}

/* p premier a 2. renvoie 1 si a est un carre dans ZK_p, 0 sinon */
static long
psquarenf(GEN nf,GEN a,GEN p)
{
  long v,ltop=avma;
  GEN norm,ap;

  if (gcmp0(a)) return 1;
  v=idealval(nf,a,p); if (v&1) return 0;
  ap=gdiv(a,gpuigs(basistoalg(nf,(GEN)p[2]),v));

  norm=gshift(idealnorm(nf,p),-1);
  ap=gmul(ap,gmodulsg(1,(GEN)p[1]));
  ap=gaddgs(gpui(ap,norm,0),-1);
  if (gcmp0(ap)) { avma=ltop; return 1; }
  ap=lift(lift(ap));
  v = idealval(nf,ap,p); avma=ltop;
  return (v>0);
}

static long
check2(GEN nf, GEN a, GEN zinit)
{
  GEN zlog=zideallog(nf,a,zinit), p1 = gmael(zinit,2,2);
  long i;

  for (i=1; i<lg(p1); i++)
    if (!mpodd((GEN)p1[i]) && mpodd((GEN)zlog[i])) return 0;
  return 1;
}

/* p divise 2. renvoie 1 si a est un carre dans ZK_p, 0 sinon */
static long
psquare2nf(GEN nf,GEN a,GEN p,GEN zinit)
{
  long v, ltop=avma;

  if (gcmp0(a)) return 1;
  v=idealval(nf,a,p); if (v&1) return 0;
  a = gdiv(a,gmodulcp(gpuigs(gmul((GEN)nf[7],(GEN)p[2]),v),(GEN)nf[1]));
  v = check2(nf,a,zinit); avma = ltop; return v;
}

/* p divise 2, (a,p)=1. renvoie 1 si a est un carre de (ZK / p^q)*, 0 sinon. */
static long
psquare2qnf(GEN nf,GEN a,GEN p,long q)
{
  long v, ltop=avma;
  GEN zinit = zidealstarinit(nf,idealpows(nf,p,q));

  v = check2(nf,a,zinit); avma = ltop; return v;
}

static long
lemma6nf(GEN nf,GEN pol,GEN p,long nu,GEN x)
{
  long i,lambda,mu,ltop=avma;
  GEN gx,gpx;

  for (i=lgef(pol)-2,gx=(GEN) pol[i+1]; i>1; i--)
    gx = gadd(gmul(gx,x),(GEN) pol[i]);
  if (psquarenf(nf,gx,p)) { avma=ltop; return 1; }
  lambda = idealval(nf,gx,p);

  for (i=lgef(pol)-2,gpx=gmulgs((GEN) pol[i+1],i-1); i>2; i--)
    gpx=gadd(gmul(gpx,x),gmulgs((GEN) pol[i],i-2));
  mu = gcmp0(gpx)? BIGINT: idealval(nf,gpx,p);

  avma=ltop;
  if (lambda > mu<<1) return 1;
  if (lambda >= nu<<1  && mu >= nu) return 0;
  return -1;
}

static long
lemma7nf(GEN nf,GEN pol,GEN p,long nu,GEN x,GEN zinit)
{
  long res,i,lambda,mu,q,ltop=avma;
  GEN gx,gpx,p1;

  for (i=lgef(pol)-2, gx=(GEN) pol[i+1]; i>1; i--)
    gx=gadd(gmul(gx,x),(GEN) pol[i]);
  if (psquare2nf(nf,gx,p,zinit)) { avma=ltop; return 1; }
  lambda=idealval(nf,gx,p);

  for (i=lgef(pol)-2,gpx=gmulgs((GEN) pol[i+1],i-1); i>2; i--)
    gpx=gadd(gmul(gpx,x),gmulgs((GEN) pol[i],i-2));
  if (!gcmp0(gpx)) mu=idealval(nf,gpx,p); else mu=BIGINT;

  if (lambda>(mu<<1)) { avma=ltop; return 1; }
  if (nu > mu)
  {
    if (lambda&1) { avma=ltop; return -1; }
    q=mu+nu-lambda; res=1;
  }
  else
  {
    if (lambda>=(nu<<1)) { avma=ltop; return 0; }
    if (lambda&1) { avma=ltop; return -1; }
    q=(nu<<1)-lambda; res=0;
  }
  if (q > itos((GEN) p[3])<<1) { avma=ltop; return -1; }
  p1 = gmodulcp(gpuigs(gmul((GEN)nf[7],(GEN)p[2]), lambda), (GEN)nf[1]);
  if (!psquare2qnf(nf,gdiv(gx,p1), p,q)) res = -1;
  avma=ltop; return res;
}

static long
zpsolnf(GEN nf,GEN pol,GEN p,long nu,GEN pnu,GEN x0,GEN repr,GEN zinit)
{
  long i,result,ltop=avma;
  GEN pnup;

  nf=checknf(nf);
  if (cmpis((GEN) p[1],2))
    result=lemma6nf(nf,pol,p,nu,x0);
  else
    result=lemma7nf(nf,pol,p,nu,x0,zinit);
  if (result== 1) return 1;
  if (result==-1) return 0;
  pnup=gmul(pnu,gmodulcp(gmul((GEN) nf[7],(GEN) p[2]),(GEN) nf[1]));
  nu++;
  for (i=1; i<lg(repr); i++)
    if (zpsolnf(nf,pol,p,nu,pnup,gadd(x0,gmul(pnu,(GEN)repr[i])),repr,zinit))
    { avma=ltop; return 1; }
  avma=ltop; return 0;
}

/* calcule un systeme de representants Zk/p */
static GEN
repres(GEN nf,GEN p)
{
  long i,j,k,f,pp,ppf,ppi;
  GEN mat,fond,rep;

  fond=cgetg(1,t_VEC);
  mat=idealhermite(nf,p);
  for (i=1; i<lg(mat); i++)
    if (!gcmp1(gmael(mat,i,i)))
      fond = concatsp(fond,gmael(nf,7,i));
  f=lg(fond)-1;
  pp=itos((GEN) p[1]);
  for (i=1,ppf=1; i<=f; i++) ppf*=pp;
  rep=cgetg(ppf+1,t_VEC);
  rep[1]=zero; ppi=1;
  for (i=0; i<f; i++,ppi*=pp)
    for (j=1; j<pp; j++)
      for (k=1; k<=ppi; k++)
	rep[j*ppi+k]=ladd((GEN) rep[k],gmulsg(j,(GEN) fond[i+1]));
  return gmodulcp(rep,(GEN) nf[1]);
}

/* =1 si l'equation y^2 = z^deg(pol) * pol(x/z) a une solution rationnelle
 *    p-adique (eventuellement (1,y,0) = oo)
 * =0 sinon.
 * Les coefficients de pol doivent etre des entiers de nf.
 * p est un ideal premier sous forme primedec.
 */
long
qpsolublenf(GEN nf,GEN pol,GEN p)
{
  GEN repr,zinit,p1;
  long ltop=avma;

  if (gcmp0(pol)) return 1;
  if (typ(pol)!=t_POL) err(notpoler,"qpsolublenf");
  if (typ(p)!=t_VEC  || lg(p)!=6)
    err(talker,"not a prime ideal in qpsolublenf");
  nf=checknf(nf);

  if (cmpis((GEN) p[1],2))
  {
    if (psquarenf(nf,(GEN) pol[2],p)) return 1;
    if (psquarenf(nf, leading_term(pol),p)) return 1;
    zinit=gzero;
  }
  else
  {
    zinit=zidealstarinit(nf,idealpows(nf,p,1+2*idealval(nf,gdeux,p)));
    if (psquare2nf(nf,(GEN) pol[2],p,zinit)) return 1;
    if (psquare2nf(nf, leading_term(pol),p,zinit)) return 1;
  }
  repr=repres(nf,p);
  if (zpsolnf(nf,pol,p,0,gun,gzero,repr,zinit)) { avma=ltop; return 1; }
  p1 = gmodulcp(gmul((GEN) nf[7],(GEN) p[2]),(GEN) nf[1]);
  if (zpsolnf(nf,polrecip(pol),p,1,p1,gzero,repr,zinit))
    { avma=ltop; return 1; }

  avma=ltop; return 0;
}

/* =1 si l'equation y^2 = pol(x) a une solution entiere p-adique
 * =0 sinon.
 * Les coefficients de pol doivent etre des entiers de nf.
 * p est un ideal premier sous forme primedec.
 */
long
zpsolublenf(GEN nf,GEN pol,GEN p)
{
  GEN repr,zinit;
  long ltop=avma;

  if (gcmp0(pol)) return 1;
  if (typ(pol)!=t_POL) err(notpoler,"zpsolublenf");
  if (typ(p)!=t_VEC || lg(p)!=6)
    err(talker,"not a prime ideal in zpsolublenf");
  nf=checknf(nf);

  if (cmpis((GEN)p[1],2))
  {
    if (psquarenf(nf,(GEN) pol[2],p)) return 1;
    zinit=gzero;
  }
  else
  {
    zinit=zidealstarinit(nf,idealpows(nf,p,1+2*idealval(nf,gdeux,p)));
    if (psquare2nf(nf,(GEN) pol[2],p,zinit)) return 1;
  }
  repr=repres(nf,p);
  if (zpsolnf(nf,pol,p,0,gun,gzero,repr,zinit)) { avma=ltop; return 1; }
  avma=ltop; return 0;
}

static long
hilb2nf(GEN nf,GEN a,GEN b,GEN p)
{
  GEN pol;
  long ltop=avma;

  a=lift(a); b=lift(b);
  pol=polx[0]; pol=gadd(gmul(a,gsqr(pol)),b);
  if (qpsolublenf(nf,pol,p)) { avma=ltop; return 1; }
  avma=ltop; return -1;
}

/* pr doit etre sous la forme primedec */
static GEN
nfmodid2(GEN nf,GEN x,GEN pr)
{
  x=lift(x);
  x=gmod(x,lift(basistoalg(nf,(GEN)pr[2])));
  return gmul(x,gmodulsg(1,(GEN)pr[1]));
}

long
nfhilbertp(GEN nf,GEN a,GEN b,GEN p)
/* calcule le symbole de Hilbert quadratique local (a,b)_p
 * en l'ideal premier p du corps nf,
 * a et b sont des elements non nuls de nf, sous la forme
 * de polymods ou de polynomes, et p renvoye par primedec.
 */
{
  GEN aux,aux2;
  long ta=typ(a),tb=typ(b),alpha,beta,sign,rep,ltop=avma;

  if ((ta!=t_INT && ta!=t_POLMOD && ta!=t_POL)
   || (tb!=t_INT && tb!=t_POLMOD && tb!=t_POL))
    err (typeer,"nfhilbertp");
  if (gcmp0(a) || gcmp0(b))
    err (talker,"0 argument in nfhilbertp");
  checkprimeid(p); nf=checknf(nf);

  if (!cmpis((GEN) p[1],2)) return hilb2nf(nf,a,b,p);

  if (ta != t_POLMOD) a=gmodulcp(a,(GEN)nf[1]);
  if (tb != t_POLMOD) b=gmodulcp(b,(GEN)nf[1]);

  alpha=idealval(nf,a,p); beta=idealval(nf,b,p);
  if ((alpha&1) == 0 && (beta&1) == 0) { avma=ltop; return 1; }
  aux2=shifti(idealnorm(nf,p),-1);
  if (alpha&1 && beta&1 && mpodd(aux2)) sign=1; else sign=-1;
  aux=nfmodid2(nf,gdiv(gpuigs(a,beta),gpuigs(b,alpha)),p); /* ?????? */
  aux=gaddgs(powgi(aux,aux2),sign);
  aux=lift(lift(aux));
  if (gcmp0(aux)) rep=1; else rep=(idealval(nf,aux,p)>=1);
  avma=ltop; return rep? 1: -1;
}

/* calcule le symbole de Hilbert quadratique global (a,b):
 * = 1 si l'equation X^2-aY^2-bZ^2=0 a une solution non triviale,
 * =-1 sinon,
 * a et b doivent etre non nuls.
 */
long
nfhilbert(GEN nf,GEN a,GEN b)
{
  long ta=typ(a),tb=typ(b),r1,i,ltop=avma;
  GEN S,al,bl;

  nf=checknf(nf);
  if ((ta!=t_INT && ta!=t_POLMOD && ta!=t_POL)
   || (tb!=t_INT && tb!=t_POLMOD && tb!=t_POL))
    err (typeer,"nfhilbert");
  if (gcmp0(a) || gcmp0(b))
    err (talker,"0 argument in nfhilbert");

  al=lift(a); bl=lift(b);
 /* solutions locales aux places infinies reelles */
  r1=itos(gmael(nf,2,1));
  for (i=1; i<=r1; i++)
    if (signe(poleval(al,gmael(nf,6,i))) < 0 &&
        signe(poleval(bl,gmael(nf,6,i))) < 0)
    {
      if (DEBUGLEVEL>=4)
      {
        fprintferr("nfhilbert not soluble at a real place\n");
        flusherr();
      }
      avma=ltop; return -1;
    }
  if (ta!=t_POLMOD) a=gmodulcp(a,(GEN)nf[1]);
  if (tb!=t_POLMOD) b=gmodulcp(b,(GEN)nf[1]);

  /* solutions locales aux places finies (celles qui divisent 2ab)*/

  S=(GEN) idealfactor(nf,gmul(gmulsg(2,a),b))[1];
  /* formule du produit ==> on peut eviter un premier */
  for (i=lg(S)-1; i>1; i--)
    if (nfhilbertp(nf,a,b,(GEN) S[i])==-1)
    {
      if (DEBUGLEVEL >=4)
      {
	fprintferr("nfhilbert not soluble at finite place: ");
	outerr((GEN)S[i]); flusherr();
      }
      avma=ltop; return -1;
    }
  avma=ltop; return 1;
}

long
nfhilbert0(GEN nf,GEN a,GEN b,GEN p)
{
  if (p) return nfhilbertp(nf,a,b,p);
  return nfhilbert(nf,a,b);
}

GEN vconcat(GEN Q1, GEN Q2);
GEN mathnfspec(GEN x, GEN *ptperm, GEN *ptdep, GEN *ptB, GEN *ptC);
/* S a list of prime ideal in primedec format. Return res:
 * res[1] = generators of (S-units / units), as polynomials
 * res[2] = [perm, HB, den], for bnfissunit
 * res[3] = [] (was: log. embeddings of res[1])
 * res[4] = S-regulator ( = R * det(res[2]) * \prod log(Norm(S[i])))
 * res[5] = S class group
 * res[6] = S
 */
GEN
bnfsunit(GEN bnf,GEN S,long prec)
{
  long i,j,ls,ltop=avma,lbot;
  GEN p1,nf,classgp,gen,M,U,H;
  GEN sunit,card,sreg,res,pow,fa = cgetg(3, t_MAT);

  if (typ(S) != t_VEC) err(typeer,"bnfsunit");
  bnf = checkbnf(bnf); nf=(GEN)bnf[7];
  classgp=gmael(bnf,8,1);
  gen = (GEN)classgp[3];

  sreg = gmael(bnf,8,2);
  res=cgetg(7,t_VEC);
  res[1]=res[2]=res[3]=lgetg(1,t_VEC);
  res[4]=(long)sreg;
  res[5]=(long)classgp;
  res[6]=(long)S; ls=lg(S);

 /* M = relation matrix for the S class group (in terms of the class group
  * generators given by gen)
  * 1) ideals in S
  */
  M = cgetg(ls,t_MAT);
  for (i=1; i<ls; i++)
  {
    p1 = (GEN)S[i]; checkprimeid(p1);
    M[i] = (long)isprincipal(bnf,p1);
  }
  /* 2) relations from bnf class group */		
  M = concatsp(M, diagonal((GEN) classgp[2]));

  /* S class group */
  H = hnfall(M); U = (GEN)H[2]; H= (GEN)H[1];
  card = gun;
  if (lg(H) > 1)
  { /* non trivial (rare!) */
    GEN SNF, ClS = cgetg(4,t_VEC);

    SNF = smith2(H); p1 = (GEN)SNF[3];
    card = dethnf_i(p1);
    ClS[1] = (long)card; /* h */
    for(i=1; i<lg(p1); i++)
      if (gcmp1((GEN)p1[i])) break;
    setlg(p1,i);
    ClS[2]=(long)p1; /* cyc */

    p1=cgetg(i,t_VEC); pow=invmat((GEN)SNF[1]);
    fa[1] = (long)gen;
    for(i--; i; i--)
    {
      fa[2] = pow[i];
      p1[i] = (long)factorback(fa, nf);
    }
    ClS[3]=(long)p1; /* gen */
    res[5]=(long) ClS;
  }

  /* S-units */
  if (ls>1)
  {
    GEN den, Sperm, perm, dep, B, U1 = U;
    long lH, lB;

   /* U1 = upper left corner of U, invertible. S * U1 = principal ideals
    * whose generators generate the S-units */
    setlg(U1,ls); p1 = cgetg(ls, t_MAT); /* p1 is junk for mathnfspec */
    for (i=1; i<ls; i++) { setlg(U1[i],ls); p1[i] = lgetg(1,t_COL); }
    H = mathnfspec(U1,&perm,&dep,&B,&p1);
    lH = lg(H);
    lB = lg(B);
    if (lg(dep) > 1 && lg(dep[1]) > 1) err(bugparier,"bnfsunit");
   /*                   [ H B  ]            [ H^-1   - H^-1 B ]
    * perm o HNF(U1) =  [ 0 Id ], inverse = [  0         Id   ]
    * (permute the rows)
    * S * HNF(U1) = _integral_ generators for S-units  = sunit */
    Sperm = cgetg(ls, t_VEC); sunit = cgetg(ls, t_VEC);
    for (i=1; i<ls; i++) Sperm[i] = S[perm[i]]; /* S o perm */

    setlg(Sperm, lH); fa[1] = (long)Sperm;
    for (i=1; i<lH; i++)
    {
      fa[2] = H[i];
      sunit[i] = (long)factorback(fa, nf);
    }
    for (i=1; i<lB; i++)
    {
      fa[2] = B[i]; j = lH-1 + i;
      sunit[j] = (long)idealmul(nf, (GEN)Sperm[j], factorback(fa, nf));
    }
    for (i=1; i<ls; i++)
      sunit[i] = isprincipalgenforce(bnf, (GEN)sunit[i])[2];

    p1 = cgetg(4,t_VEC);
    den = dethnf_i(H); H = gmul(den, invmat(H)); 
    p1[1] = (long)perm;
    p1[2] = (long)concatsp(H, gneg(gmul(H,B))); /* top part of inverse * den */
    p1[3] = (long)den; /* keep denominator separately */
    sunit = basistoalg(nf,sunit);
    res[2] = (long)p1; /* HNF in split form perm + (H B) [0 Id missing] */
    res[1] = (long)lift_intern(sunit);
  }

  /* S-regulator */
  sreg = gmul(sreg,card);
  for (i=1; i<ls; i++)
  {
    GEN p = (GEN)S[i];
    if (typ(p) == t_VEC) p = (GEN) p[1];
    sreg = gmul(sreg,glog(p,prec));
  }
  res[4]=(long) sreg; lbot=avma;
  return gerepile(ltop,lbot,gcopy(res));
}

/* cette fonction est l'equivalent de isunit, sauf qu'elle donne le resultat
 * avec des s-unites: si x n'est pas une s-unite alors issunit=[]~;
 * si x est une s-unite alors
 * x=\prod_{i=0}^r {e_i^issunit[i]}*prod{i=r+1}^{r+s} {s_i^issunit[i]}
 * ou les e_i sont les unites du corps (comme dans isunit)
 * et les s_i sont les s-unites calculees par sunit (dans le meme ordre).
 */
GEN
bnfissunit(GEN bnf,GEN suni,GEN x)
{
  long lB,cH,i,k,ls,tetpil, av = avma;
  GEN den,gen,S,v,p1,xp,xm,xb,N,HB,perm;

  bnf = checkbnf(bnf);
  if (typ(suni)!=t_VEC || lg(suni)!=7) err(typeer,"bnfissunit");
  switch (typ(x))
  {
    case t_INT: case t_FRAC: case t_FRACN:
    case t_POL: case t_COL:
      x = basistoalg(bnf,x); break;
    case t_POLMOD: break;
    default: err(typeer,"bnfissunit");
  }
  if (gcmp0(x)) return cgetg(1,t_COL);

  S = (GEN) suni[6]; ls=lg(S);
  if (ls==1) return isunit(bnf,x);

  p1 = (GEN)suni[2];
  perm = (GEN)p1[1];
  HB = (GEN)p1[2]; den = (GEN)p1[3];
  cH = lg(HB[1]) - 1;
  lB = lg(HB) - cH;
  xb = algtobasis(bnf,x); p1 = denom(content(xb));
  N = mulii(gnorm(gmul(x,p1)), p1); /* relevant primes divide N */
  v = cgetg(ls, t_VECSMALL);
  for (i=1; i<ls; i++)
  {
    GEN P = (GEN)S[i];
    v[i] = (resii(N, (GEN)P[1]) == gzero)? element_val(bnf,xb,P): 0;
  }
  /* here, x = S v */
  p1 = cgetg(ls, t_COL);
  for (i=1; i<ls; i++) p1[i] = lstoi(v[perm[i]]); /* p1 = v o perm */
  v = gmul(HB, p1);
  for (i=1; i<=cH; i++)
  {
    GEN w = gdiv((GEN)v[i], den);
    if (typ(w) != t_INT) { avma = av; return cgetg(1,t_COL); }
    v[i] = (long)w;
  }
  p1 += cH;
  p1[0] = evaltyp(t_COL) | evallg(lB);
  v = concatsp(v, p1); /* append bottom of p1 (= [0 Id] part) */

  xp = gun; xm = gun; gen = (GEN)suni[1];
  for (i=1; i<ls; i++)
  {
    k = -itos((GEN)v[i]); if (!k) continue;
    p1 = basistoalg(bnf, (GEN)gen[i]);
    if (k > 0) xp = gmul(xp, gpuigs(p1, k));
    else       xm = gmul(xm, gpuigs(p1,-k));
  }
  if (xp != gun) x = gmul(x,xp);
  if (xm != gun) x = gdiv(x,xm);
  p1 = isunit(bnf,x);
  if (lg(p1)==1) { avma = av; return cgetg(1,t_COL); }
  tetpil=avma; return gerepile(av,tetpil,concat(p1,v));
}

static void
vecconcat(GEN bnf,GEN relnf,GEN vec,GEN *prod,GEN *S1,GEN *S2)
{
  long i;

  for (i=1; i<lg(vec); i++)
    if (signe(resii(*prod,(GEN)vec[i])))
    {
       *prod=mulii(*prod,(GEN)vec[i]);
       *S1=concatsp(*S1,primedec(bnf,(GEN)vec[i]));
       *S2=concatsp(*S2,primedec(relnf,(GEN)vec[i]));
    }
}

/* bnf est le corps de base (buchinitfu).
 * ext definit l'extension relative:
 * ext[1] est une equation relative du corps,
 * telle qu'une de ses racines engendre le corps sur Q.
 * ext[2] exprime le generateur (y) du corps de base,
 * en fonction de la racine (x) de ext[1],
 * ext[3] est le buchinitfu (sur Q) de l'extension.

 * si flag=0 c'est qu'on sait a l'avance que l'extension est galoisienne,
 * et dans ce cas la reponse est exacte.
 * si flag>0 alors on ajoue dans S tous les ideaux qui divisent p<=flag.
 * si flag<0 alors on ajoute dans S tous les ideaux qui divisent -flag.

 * la reponse est un vecteur v a 2 composantes telles que
 * x=N(v[1])*v[2].
 * x est une norme ssi v[2]=1.
 */
GEN
rnfisnorm(GEN bnf,GEN ext,GEN x,long flag,long PREC)
{
  long lgsunitrelnf,i,lbot, ltop = avma;
  GEN relnf,aux,vec,tors,xnf,H,Y,M,A,suni,sunitrelnf,sunitnormnf,prod;
  GEN res = cgetg(3,t_VEC), S1,S2;

  if (typ(ext)!=t_VEC || lg(ext)!=4) err (typeer,"bnfisnorm");
  bnf = checkbnf(bnf); relnf = (GEN)ext[3];
  if (gcmp0(x) || gcmp1(x) || (gcmp_1(x) && (degree((GEN)ext[1])&1)))
  {
    res[1]=lcopy(x); res[2]=un; return res;
  }

/* construction de l'ensemble S des ideaux
   qui interviennent dans les solutions */

  prod=gun; S1=S2=cgetg(1,t_VEC);
  if (!gcmp1(gmael3(relnf,8,1,1)))
  {
    GEN genclass=gmael3(relnf,8,1,3);
    vec=cgetg(1,t_VEC);
    for(i=1;i<lg(genclass);i++)
      if (!gcmp1(ggcd(gmael4(relnf,8,1,2,i), stoi(degree((GEN)ext[1])))))
        vec=concatsp(vec,(GEN)factor(gmael3(genclass,i,1,1))[1]);
    vecconcat(bnf,relnf,vec,&prod,&S1,&S2);
  }

  if (flag>1)
  {
    for (i=2; i<=flag; i++)
      if (isprime(stoi(i)) && signe(resis(prod,i)))
      {
	prod=mulis(prod,i);
	S1=concatsp(S1,primedec(bnf,stoi(i)));
	S2=concatsp(S2,primedec(relnf,stoi(i)));
      }
  }
  else if (flag<0)
    vecconcat(bnf,relnf,(GEN)factor(stoi(-flag))[1],&prod,&S1,&S2);

  if (flag)
  {
    GEN normdiscrel=divii(gmael(relnf,7,3),
			  gpuigs(gmael(bnf,7,3),lg(ext[1])-3));
    vecconcat(bnf,relnf,(GEN) factor(absi(normdiscrel))[1],
	      &prod,&S1,&S2);
  }
  vec=(GEN) idealfactor(bnf,x)[1]; aux=cgetg(2,t_VEC);
  for (i=1; i<lg(vec); i++)
    if (signe(resii(prod,gmael(vec,i,1))))
    {
      aux[1]=vec[i]; S1=concatsp(S1,aux);
    }
  xnf=lift(x);
  xnf=gsubst(xnf,varn(xnf),(GEN)ext[2]);
  vec=(GEN) idealfactor(relnf,xnf)[1];
  for (i=1; i<lg(vec); i++)
    if (signe(resii(prod,gmael(vec,i,1))))
    {
      aux[1]=vec[i]; S2=concatsp(S2,aux);
    }

  res[1]=un; res[2]=(long)x;
  tors=cgetg(2,t_VEC); tors[1]=mael3(relnf,8,4,2);

  /* calcul sur les S-unites */

  suni=bnfsunit(bnf,S1,PREC);
  A=lift(bnfissunit(bnf,suni,x));
  sunitrelnf=(GEN) bnfsunit(relnf,S2,PREC)[1];
  if (lg(sunitrelnf)>1)
  {
    sunitrelnf=lift(basistoalg(relnf,sunitrelnf));
    sunitrelnf=concatsp(tors,sunitrelnf);
  }
  else sunitrelnf=tors;
  aux=(GEN)relnf[8];
  if (lg(aux)>=6) aux=(GEN)aux[5];
  else
  {
    aux=buchfu(relnf);
    if(gcmp0((GEN)aux[2]))
      err(talker,"please increase precision to have units in bnfisnorm");
    aux=(GEN)aux[1];
  }
  if (lg(aux)>1)
    sunitrelnf=concatsp(gtrans(aux),sunitrelnf);
  lgsunitrelnf=lg(sunitrelnf);
  M=cgetg(lgsunitrelnf+1,t_MAT);
  sunitnormnf=cgetg(lgsunitrelnf,t_VEC);
  for (i=1; i<lgsunitrelnf; i++)
  {
    sunitnormnf[i]=lnorm(gmodulcp((GEN) sunitrelnf[i],(GEN)ext[1]));
    M[i]=llift(bnfissunit(bnf,suni,(GEN) sunitnormnf[i]));
  }
  M[lgsunitrelnf]=lgetg(lg(A),t_COL);
  for (i=1; i<lg(A); i++) mael(M,lgsunitrelnf,i)=zero;
  mael(M,lgsunitrelnf,lg(mael(bnf,7,6))-1)=mael3(bnf,8,4,1);
  H=hnfall(M); Y=inverseimage(gmul(M,(GEN) H[2]),A);
  Y=gmul((GEN) H[2],Y);
  for (aux=(GEN)res[1],i=1; i<lgsunitrelnf; i++)
    aux=gmul(aux,gpuigs(gmodulcp((GEN) sunitrelnf[i],(GEN)ext[1]),
                        itos(gfloor((GEN)Y[i]))));
  res[1]=(long)aux;
  res[2]=ldiv(x,gnorm(gmodulcp(lift(aux),(GEN)ext[1])));

  lbot=avma; return gerepile(ltop,lbot,gcopy(res));
}

GEN
bnfisnorm(GEN bnf,GEN x,long flag,long PREC)
{
  long ltop = avma, lbot;
  GEN ext = cgetg(4,t_VEC);

  bnf = checkbnf(bnf);
  ext[1] = mael(bnf,7,1);
  ext[2] = zero;
  ext[3] = (long) bnf;
  bnf = buchinitfu(polx[MAXVARN],NULL,NULL,0); lbot = avma;
  return gerepile(ltop,lbot,rnfisnorm(bnf,ext,x,flag,PREC));
}