GEN
nffactormod(GEN nf, GEN x, GEN pr)
{
long j, l, vx = varn(x), vn;
pari_sp av = avma;
GEN F, E, rep, xrd, modpr, T, p;
nf = checknf(nf);
vn = varn(nf[1]);
if (typ(x)!=t_POL) pari_err(typeer,"nffactormod");
if (varncmp(vx,vn) >= 0)
pari_err(talker,"polynomial variable must have highest priority in nffactormod");
modpr = nf_to_ff_init(nf, &pr, &T, &p);
xrd = modprX(x, nf, modpr);
rep = FqX_factor(xrd,T,p);
settyp(rep, t_MAT);
F = gel(rep,1); l = lg(F);
E = gel(rep,2); settyp(E, t_COL);
for (j = 1; j < l; j++) {
gel(F,j) = modprX_lift(gel(F,j), modpr);
gel(E,j) = stoi(E[j]);
}
return gerepilecopy(av, rep);
}
GEN
bnflogdegree(GEN nf, GEN A, GEN ell)
{
pari_sp av = avma;
GEN AZ, A0Z, NA0;
long vAZ;
if (typ(ell) != t_INT) pari_err_TYPE("bnflogdegree", ell);
nf = checknf(nf);
A = idealhnf(nf, A);
AZ = gcoeff(A,1,1);
vAZ = Z_pvalrem(AZ, ell, &A0Z);
if (is_pm1(A0Z))
NA0 = gen_1;
else
(void)Z_pvalrem(idealnorm(nf,A), ell, &NA0);
if (vAZ)
{
GEN Aell = ZM_hnfmodid(A, powiu(ell,vAZ));
GEN S = idealprimedec(nf, ell), T;
long l, i, s = 0;
T = padicfact(nf, S, 100);
l = lg(S);
for (i = 1; i < l; i++)
{
GEN P = gel(S,i);
long v = idealval(nf, Aell, P);
if (v) s += v * ftilde(nf, P, gel(T,i));
}
if (s) NA0 = gmul(NA0, gpowgs(ell1(ell), s));
}
return gerepileupto(av, NA0);
}
/* K number field, pr a maximal ideal, let K_pr be the attached local
* field, K_pr = Q_p[X] / (T), T irreducible. Return \tilde{e}(K_pr/Q_p) */
static long
etilde(GEN nf, GEN pr, GEN T)
{
GEN gp = pr_get_p(pr);
ulong e = pr_get_e(pr);
long v, voo, vmin, p, k;
if (!u_pval(e, gp))
{
v = u_pval(pr_get_f(pr), gp);
return itou( mului(e, powiu(gp, v)) );
}
nf = checknf(nf);
p = itou(gp);
k = e / (p-1) + 1;
/* log Norm_{F_P/Q_p} (1 + P^k) = Tr(P^k) = p^[(k + v(Diff))/ e] Z_p */
voo = (k + idealval(nf, nf_get_diff(nf), pr)) / e;
vmin = vlognorm(nf, T, pr_get_gen(pr), gp, voo);
if (k > 1)
{
GEN U = idealprincipalunits(nf, pr, k);
GEN gen = abgrp_get_gen(U), cyc = abgrp_get_cyc(U);
long i, l = lg(cyc);
for (i = 1; i < l; i++)
{
if (voo - Z_lval(gel(cyc,i), p) >= vmin) break;
vmin = vlognorm(nf, T, gel(gen,i), gp, vmin);
}
}
v = u_lval(degpol(T), p) + (p == 2UL? 2 : 1) - vmin;
(void)u_lvalrem(e, p, &e);
return e * upowuu(p,v);
}
/* assume x is squarefree */
int
nfissplit(GEN nf, GEN x)
{
pari_sp av = avma;
long l;
if (typ(x) != t_POL) pari_err(typeer, "nfissplit");
l = lg(nfsqff(checknf(nf), x, 2));
avma = av; return l != 1;
}
GEN
nfrootsall_and_pr(GEN nf, GEN pol)
{
GEN J1,J2, pr, T;
pari_sp av = avma;
GEN z = gerepileupto(av, nfsqff(checknf(nf), pol, 2));
if (lg(z) == 1) return NULL;
(void)nf_pick_prime(1, nf, unifpol(nf, pol, t_COL), 2,
&J1, &J2, &pr, &T);
return mkvec2(z, pr);
}
GEN
bnflogef(GEN nf, GEN pr)
{
pari_sp av = avma;
long e, f, ef;
GEN p;
checkprid(pr); p = pr_get_p(pr);
nf = checknf(nf);
e = pr_get_e(pr);
f = pr_get_f(pr); ef = e*f;
if (u_pval(ef, p))
{
GEN T = gel(factorpadic(nf_get_pol(nf), p, 100), 1);
long j = get_ZpX_index(nf, pr, T);
e = etilde(nf, pr, gel(T,j));
f = ef / e;
}
avma = av; return mkvec2s(e,f);
}
/* return the characteristic polynomial of alpha over nf, where alpha
is an element of the algebra nf[X]/(T) given as a polynomial in X */
GEN
rnfcharpoly(GEN nf, GEN T, GEN alpha, long v)
{
long vnf, vT, lT;
pari_sp av = avma;
GEN p1;
nf=checknf(nf); vnf = varn(nf[1]);
if (v<0) v = 0;
T = fix_relative_pol(nf,T,1);
if (typ(alpha) == t_POLMOD) alpha = lift_to_pol(alpha);
lT = lg(T);
if (typ(alpha) != t_POL || varn(alpha) == vnf)
return gerepileupto(av, gpowgs(gsub(pol_x[v], alpha), lT - 3));
vT = varn(T);
if (varn(alpha) != vT || varncmp(v, vnf)>=0)
pari_err(talker,"incorrect variables in rnfcharpoly");
if (lg(alpha) >= lT) alpha = RgX_rem(alpha, T);
if (lT <= 4)
return gerepileupto(av, gsub(pol_x[v], alpha));
p1 = caract2(T, unifpol(nf,alpha, t_POLMOD), v);
return gerepileupto(av, unifpol(nf, p1, t_POLMOD));
}
/* return the roots of pol in nf */
GEN
nfroots(GEN nf,GEN pol)
{
pari_sp av = avma;
GEN A,g, T;
long d;
if (!nf) return nfrootsQ(pol);
nf = checknf(nf); T = gel(nf,1);
if (typ(pol) != t_POL) pari_err(notpoler,"nfroots");
if (varncmp(varn(pol), varn(T)) >= 0)
pari_err(talker,"polynomial variable must have highest priority in nfroots");
d = degpol(pol);
if (d == 0) return cgetg(1,t_VEC);
if (d == 1)
{
A = gneg_i(gdiv(gel(pol,2),gel(pol,3)));
return gerepilecopy(av, mkvec( basistoalg(nf,A) ));
}
A = fix_relative_pol(nf,pol,0);
A = Q_primpart( lift_intern(A) );
if (DEBUGLEVEL>3) fprintferr("test if polynomial is square-free\n");
g = nfgcd(A, derivpol(A), T, gel(nf,4));
if (degpol(g))
{ /* not squarefree */
g = QXQX_normalize(g, T);
A = RgXQX_div(A,g,T);
}
A = QXQX_normalize(A, T);
A = Q_primpart(A);
A = nfsqff(nf,A,1);
A = RgXQV_to_mod(A, T);
return gerepileupto(av, gen_sort(A, 0, cmp_pol));
}
/* return the factorization of x in nf */
GEN
nffactor(GEN nf,GEN pol)
{
GEN A,g,y,p1,T, rep = cgetg(3, t_MAT);
long l, j, dA;
pari_sp av = avma;
pari_timer ti;
if (DEBUGLEVEL>2) { TIMERstart(&ti); fprintferr("\nEntering nffactor:\n"); }
nf = checknf(nf); T = gel(nf,1);
if (typ(pol) != t_POL) pari_err(notpoler,"nffactor");
if (varncmp(varn(pol), varn(T)) >= 0)
pari_err(talker,"polynomial variable must have highest priority in nffactor");
A = fix_relative_pol(nf,pol,0);
dA = degpol(A);
if (dA <= 0) {
avma = (pari_sp)(rep + 3);
return dA == 0? trivfact(): zerofact(varn(pol));
}
A = Q_primpart( QXQX_normalize(A, T) );
if (dA == 1) {
GEN c;
A = gerepilecopy(av, A); c = gel(A,2);
if (typ(c) == t_POL && degpol(c) > 0) gel(A,2) = mkpolmod(c, gcopy(T));
gel(rep,1) = mkcol(A);
gel(rep,2) = mkcol(gen_1); return rep;
}
if (degpol(T) == 1)
return gerepileupto(av, factpol(Q_primpart(simplify(pol)), 0));
A = Q_primpart( lift_intern(A) );
g = nfgcd(A, derivpol(A), T, gel(nf,4));
A = QXQX_normalize(A, T);
A = Q_primpart(A);
if (DEBUGLEVEL>2) msgTIMER(&ti, "squarefree test");
if (degpol(g))
{ /* not squarefree */
pari_sp av1;
GEN ex;
g = QXQX_normalize(g, T);
A = RgXQX_div(A,g, T);
y = nfsqff(nf,A,0); av1 = avma;
l = lg(y);
ex=(GEN)gpmalloc(l * sizeof(long));
for (j=l-1; j>=1; j--)
{
GEN fact = lift(gel(y,j)), quo = g, q;
long e = 0;
for(e = 1;; e++)
{
q = RgXQX_divrem(quo,fact,T, ONLY_DIVIDES);
if (!q) break;
quo = q;
}
ex[j] = e;
}
avma = av1; y = gerepileupto(av, RgXQXV_to_mod(y,T));
p1 = cgetg(l, t_COL); for (j=l-1; j>=1; j--) gel(p1,j) = utoipos(ex[j]);
free(ex);
}
else
{
y = gerepileupto(av, RgXQXV_to_mod(nfsqff(nf,A,0), T));
l = lg(y);
p1 = cgetg(l, t_COL); for (j=l-1; j>=1; j--) gel(p1,j) = gen_1;
}
if (DEBUGLEVEL>3)
fprintferr("number of factor(s) found: %ld\n", lg(y)-1);
gel(rep,1) = y;
gel(rep,2) = p1; return sort_factor(rep, cmp_pol);
}
static GEN
bnflog_i(GEN bnf, GEN ell)
{
long prec0, prec;
GEN nf, US, vdegS, S, T, M, CLp, CLt, Ftilde, vtG, ellk;
GEN D, Ap, cycAp, bnfS;
long i, j, lS, lvAp;
checkbnf(bnf);
nf = checknf(bnf);
S = idealprimedec(nf, ell);
bnfS = bnfsunit0(bnf, S, nf_GENMAT, LOWDEFAULTPREC); /* S-units */
US = leafcopy(gel(bnfS,1));
prec0 = maxss(30, vtilde_prec(nf, US, ell));
US = shallowconcat(bnf_get_fu(bnf), US);
settyp(US, t_COL);
T = padicfact(nf, S, prec0);
lS = lg(S); Ftilde = cgetg(lS, t_VECSMALL);
for (j = 1; j < lS; j++) Ftilde[j] = ftilde(nf, gel(S,j), gel(T,j));
CLp = CL_prime(bnf, ell, S);
cycAp = gel(CLp,1);
Ap = gel(CLp,2);
for(;;)
{
CLt = CL_tilde(nf, US, ell, T, Ftilde, &vtG, prec0);
if (CLt) break;
prec0 <<= 1;
T = padicfact(nf, S, prec0);
}
prec = ellexpo(cycAp, ell) + ellexpo(CLt,ell) + 1;
if (prec == 1) return mkvec3(cgetg(1,t_VEC), cgetg(1,t_VEC), cgetg(1,t_VEC));
vdegS = get_vdegS(Ftilde, ell, prec0);
ellk = powiu(ell, prec);
lvAp = lg(Ap);
if (lvAp > 1)
{
GEN Kcyc = bnf_get_cyc(bnf);
GEN C = zeromatcopy(lvAp-1, lS-1);
GEN Rell = gel(CLp,3), Uell = gel(CLp,4), ordS = gel(CLp,5);
for (i = 1; i < lvAp; i++)
{
GEN a, b, bi, A = gel(Ap,i), d = gel(cycAp,i);
bi = isprincipal(bnf, A);
a = vecmodii(ZC_Z_mul(bi,d), Kcyc);
/* a in subgroup generated by S = Rell; hence b integral */
b = hnf_invimage(Rell, a);
b = vecmodii(ZM_ZC_mul(Uell, ZC_neg(b)), ordS);
A = mkvec2(A, cgetg(1,t_MAT));
A = idealpowred(nf, A, d);
/* find a principal representative of A_i^cycA_i up to elements of S */
a = isprincipalfact(bnf,gel(A,1),S,b,nf_GENMAT|nf_FORCE);
if (!gequal0(gel(a,1))) pari_err_BUG("bnflog");
a = famat_mul_shallow(gel(A,2), gel(a,2)); /* principal part */
if (lg(a) == 1) continue;
for (j = 1; j < lS; j++)
gcoeff(C,i,j) = vtilde(nf, a, gel(T,j), gel(vdegS,j), ell, prec0);
}
C = gmod(gneg(C),ellk);
C = shallowtrans(C);
M = mkmat2(mkcol2(diagonal_shallow(cycAp), C), mkcol2(gen_0, vtG));
M = shallowmatconcat(M); /* relation matrix */
}
else
M = vtG;
M = ZM_hnfmodid(M, ellk);
D = matsnf0(M, 4);
if (lg(D) == 1 || !dvdii(gel(D,1), ellk))
pari_err_BUG("bnflog [missing Z_l component]");
D = vecslice(D,2,lg(D)-1);
return mkvec3(D, CLt, ellsylow(cycAp, ell));
}
