/* 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);
}
static void
bestlift_init(long a, GEN nf, GEN pr, GEN C, nflift_t *L)
{
const long D = 100;
const double alpha = ((double)D-1) / D; /* LLL parameter */
const long d = degpol(nf[1]);
pari_sp av = avma;
GEN prk, PRK, B, GSmin, pk;
pari_timer ti;
TIMERstart(&ti);
if (!a) a = (long)bestlift_bound(C, d, alpha, pr_norm(pr));
for (;; avma = av, a<<=1)
{
if (DEBUGLEVEL>2) fprintferr("exponent: %ld\n",a);
PRK = prk = idealpows(nf, pr, a);
pk = gcoeff(prk,1,1);
/* reduce size first, "scramble" matrix */
PRK = lllintpartial_ip(PRK);
/* now floating point reduction is fast */
PRK = lllint_fp_ip(PRK, 4);
PRK = lllint_i(PRK, D, 0, NULL, NULL, &B);
if (!PRK) { PRK = prk; GSmin = pk; } /* nf = Q */
else
{
pari_sp av2 = avma;
GEN S = invmat( get_R(PRK) ), BB = GS_norms(B, DEFAULTPREC);
GEN smax = gen_0;
long i, j;
for (i=1; i<=d; i++)
{
GEN s = gen_0;
for (j=1; j<=d; j++)
s = gadd(s, gdiv( gsqr(gcoeff(S,i,j)), gel(BB,j)));
if (gcmp(s, smax) > 0) smax = s;
}
GSmin = gerepileupto(av2, ginv(gmul2n(smax, 2)));
}
if (gcmp(GSmin, C) >= 0) break;
}
if (DEBUGLEVEL>2)
fprintferr("for this exponent, GSmin = %Z\nTime reduction: %ld\n",
GSmin, TIMER(&ti));
L->k = a;
L->den = L->pk = pk;
L->prk = PRK;
L->iprk = ZM_inv(PRK, pk);
L->GSmin= GSmin;
L->prkHNF = prk;
init_proj(L, gel(nf,1), gel(pr,1));
}
