/* K a bnf, compute Cl'(K) = ell-Sylow of Cl(K) / (places above ell).
* Return [D, u, R0, U0, ordS]
* - D: cyclic factors for Cl'(K)
* - u: generators of cyclic factors (all coprime to ell)
* - R0: subgroup isprincipal(<S>) (divides K.cyc)
* - U0: generators of R0 are of the form S . U0
* - ordS[i] = order of S[i] in CL(K) */
static GEN
CL_prime(GEN K, GEN ell, GEN Sell)
{
GEN g, ordS, R0, U0, U, D, u, cyc = bnf_get_cyc(K);
long i, l, lD, lS = lg(Sell);
g = leafcopy(bnf_get_gen(K));
l = lg(g);
for (i = 1; i < l; i++)
{
GEN A = gel(g,i), a = gcoeff(A,1,1);
long v = Z_pvalrem(a, ell, &a);
if (v) gel(g,i) = hnfmodid(A, a); /* make coprime to ell */
}
R0 = cgetg(lS, t_MAT);
ordS = cgetg(lS, t_VEC);
for (i = 1; i < lS; i++)
{
gel(R0,i) = isprincipal(K, gel(Sell,i));
gel(ordS,i) = charorder(cyc, gel(R0,i)); /* order of Sell[i] */
}
R0 = shallowconcat(R0, diagonal_shallow(cyc));
/* R0 = subgroup generated by S in Cl(K) [ divides diagonal(K.cyc) ]*/
R0 = ZM_hnfall(R0, &U0, 2); /* [S | cyc] * U0 = R0 in HNF */
D = ZM_snfall(R0, &U,NULL);
D = RgM_diagonal_shallow(D);
lD = lg(D);
u = ZM_inv(U, gen_1); settyp(u, t_VEC);
for (i = 1; i < lD; i++) gel(u,i) = idealfactorback(K,g,gel(u,i),1);
setlg(U0, l);
U0 = rowslice(U0,1,lS-1); /* restrict to 'S' part */
return mkvec5(D, u, R0, U0, ordS);
}
GEN
F2xq_ellcard(GEN a, GEN a6, GEN T)
{
pari_sp av = avma;
long n = F2x_degree(T);
GEN q = int2u(n), c;
if (typ(a)==t_VECSMALL)
{
GEN t = F2xq_elltrace_Harley(a6, T);
c = addii(q, F2xq_trace(a,T) ? addui(1,t): subui(1,t));
} else if (n==1)
{
long a4i = lgpol(gel(a,2)), a6i = lgpol(a6);
return utoi(a4i? (a6i? 1: 5): 3);
}
else if (n==2)
{
GEN a3 = gel(a,1), a4 = gel(a,2), x = polx_F2x(T[1]), x1 = pol1_F2x(T[1]);
GEN a613 = F2xq_mul(F2x_add(x1, a6),a3,T), a43= F2xq_mul(a4,a3,T);
long f0= F2xq_trace(F2xq_mul(a6,a3,T),T);
long f1= F2xq_trace(F2x_add(a43,a613),T);
long f2= F2xq_trace(F2x_add(F2xq_mul(a43,x,T),a613),T);
long f3= F2xq_trace(F2x_add(F2xq_mul(a43,F2x_add(x,x1),T),a613),T);
c = utoi(9-2*(f0+f1+f2+f3));
}
else
{
struct _F2xqE e;
long m = (n+1)>>1;
GEN q1 = addis(q, 1);
GEN v = n==4 ? mkvec4s(13,17,21,25)
: odd(n) ? mkvec3(subii(q1,int2u(m)),q1,addii(q1,int2u(m))):
mkvec5(subii(q1,int2u(m+1)),subii(q1,int2u(m)),q1,
addii(q1,int2u(m)),addii(q1,int2u(m+1)));
e.a2=a; e.a6=a6; e.T=T;
c = gen_select_order(v,(void*)&e, &F2xqE_group);
if (n==4 && equaliu(c, 21)) /* Ambiguous case */
{
GEN d = F2xq_powu(polx_F2x(T[1]),3,T), a3 = gel(a,1);
e.a6 = F2x_add(a6,F2xq_mul(d,F2xq_sqr(a3,T),T)); /* twist */
c = subui(34, gen_select_order(mkvec2s(13,25),(void*)&e, &F2xqE_group));
}
}
return gerepileuptoint(av, c);
}