Line data Source code
1 : /* Copyright (C) 2000 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 :
15 : /*******************************************************************/
16 : /* */
17 : /* BASIC NF OPERATIONS */
18 : /* (continued) */
19 : /* */
20 : /*******************************************************************/
21 : #include "pari.h"
22 : #include "paripriv.h"
23 :
24 : #define DEBUGLEVEL DEBUGLEVEL_nf
25 :
26 : /*******************************************************************/
27 : /* */
28 : /* IDEAL OPERATIONS */
29 : /* */
30 : /*******************************************************************/
31 :
32 : /* A valid ideal is either principal (valid nf_element), or prime, or a matrix
33 : * on the integer basis in HNF.
34 : * A prime ideal is of the form [p,a,e,f,b], where the ideal is p.Z_K+a.Z_K,
35 : * p is a rational prime, a belongs to Z_K, e=e(P/p), f=f(P/p), and b
36 : * is Lenstra's constant, such that p.P^(-1)= p Z_K + b Z_K.
37 : *
38 : * An extended ideal is a couple [I,F] where I is an ideal and F is either an
39 : * algebraic number, or a factorization matrix attached to an algebraic number.
40 : * All routines work with either extended ideals or ideals (an omitted F is
41 : * assumed to be factor(1)). All ideals are output in HNF form. */
42 :
43 : /* types and conversions */
44 :
45 : long
46 16075772 : idealtyp(GEN *ideal, GEN *arch)
47 : {
48 16075772 : GEN x = *ideal;
49 16075772 : long t,lx,tx = typ(x);
50 :
51 16075772 : if (tx!=t_VEC || lg(x)!=3) { if (arch) *arch = NULL; }
52 : else
53 : {
54 1239709 : GEN a = gel(x,2);
55 1239709 : if (typ(a) == t_MAT && lg(a) != 3)
56 : { /* allow [;] */
57 14 : if (lg(a) != 1) pari_err_TYPE("idealtyp [extended ideal]",x);
58 7 : if (arch) *arch = trivial_fact();
59 : }
60 : else
61 1239695 : if (arch) *arch = a;
62 1239702 : x = gel(x,1); tx = typ(x);
63 : }
64 16075765 : switch(tx)
65 : {
66 12222684 : case t_MAT: lx = lg(x);
67 12222684 : if (lx == 1) { t = id_PRINCIPAL; x = gen_0; break; }
68 12222523 : if (lx != lgcols(x)) pari_err_TYPE("idealtyp [nonsquare t_MAT]",x);
69 12222464 : t = id_MAT;
70 12222464 : break;
71 :
72 2819981 : case t_VEC:
73 2819981 : if (!checkprid_i(x)) pari_err_TYPE("idealtyp [fake prime ideal]",x);
74 2819935 : t = id_PRIME; break;
75 :
76 1033360 : case t_POL: case t_POLMOD: case t_COL:
77 : case t_INT: case t_FRAC:
78 1033360 : t = id_PRINCIPAL; break;
79 0 : default:
80 0 : pari_err_TYPE("idealtyp",x);
81 : return 0; /*LCOV_EXCL_LINE*/
82 : }
83 16075920 : *ideal = x; return t;
84 : }
85 :
86 : /* true nf; v = [a,x,...], a in Z. Return (a,x) */
87 : GEN
88 683249 : idealhnf_two(GEN nf, GEN v)
89 : {
90 683249 : GEN p = gel(v,1), pi = gel(v,2), m = zk_scalar_or_multable(nf, pi);
91 683239 : if (typ(m) == t_INT) return scalarmat(gcdii(m,p), nf_get_degree(nf));
92 610636 : return ZM_hnfmodid(m, p);
93 : }
94 : /* true nf */
95 : GEN
96 3605922 : pr_hnf(GEN nf, GEN pr)
97 : {
98 3605922 : GEN p = pr_get_p(pr), m;
99 3605896 : if (pr_is_inert(pr)) return scalarmat(p, nf_get_degree(nf));
100 3170368 : m = zk_scalar_or_multable(nf, pr_get_gen(pr));
101 3170002 : return ZM_hnfmodprime(m, p);
102 : }
103 :
104 : GEN
105 1366635 : idealhnf_principal(GEN nf, GEN x)
106 : {
107 : GEN cx;
108 1366635 : x = nf_to_scalar_or_basis(nf, x);
109 1366633 : switch(typ(x))
110 : {
111 1051893 : case t_COL: break;
112 281955 : case t_INT: if (!signe(x)) return cgetg(1,t_MAT);
113 280324 : return scalarmat(absi_shallow(x), nf_get_degree(nf));
114 32784 : case t_FRAC:
115 32784 : return scalarmat(Q_abs_shallow(x), nf_get_degree(nf));
116 1 : default: pari_err_TYPE("idealhnf",x);
117 : }
118 1051893 : x = Q_primitive_part(x, &cx);
119 1051888 : RgV_check_ZV(x, "idealhnf");
120 1051888 : x = zk_multable(nf, x);
121 1051891 : x = ZM_hnfmodid(x, zkmultable_capZ(x));
122 1051897 : return cx? ZM_Q_mul(x,cx): x;
123 : }
124 :
125 : /* x integral ideal in t_MAT form, nx columns */
126 : static GEN
127 91 : vec_mulid(GEN nf, GEN x, long nx, long N)
128 : {
129 91 : GEN m = cgetg(nx*N + 1, t_MAT);
130 : long i, j, k;
131 287 : for (i=k=1; i<=nx; i++)
132 2037 : for (j=1; j<=N; j++) gel(m, k++) = zk_ei_mul(nf, gel(x,i),j);
133 91 : return m;
134 : }
135 :
136 : static GEN
137 84 : nfV_idealhnf(GEN nf, GEN v)
138 : {
139 84 : GEN M = matalgtobasis(nf, v);
140 84 : GEN den, H = ZM_hnf(Q_remove_denom(M, &den));
141 84 : GEN x = vec_mulid(nf, H, lg(H)-1, nf_get_degree(nf));
142 84 : x = ZM_hnfmod(x, ZM_detmult(x));
143 84 : return den? ZM_Q_mul(x, den): x;
144 : }
145 :
146 : /* true nf */
147 : GEN
148 1743756 : idealhnf_shallow(GEN nf, GEN x)
149 : {
150 1743756 : long tx = typ(x), lx = lg(x), N;
151 :
152 : /* cannot use idealtyp because here we allow nonsquare matrices */
153 1743756 : if (tx == t_VEC && lx == 3) { x = gel(x,1); tx = typ(x); lx = lg(x); }
154 1743756 : if (tx == t_VEC && lx == 6)
155 : {
156 483951 : if (!checkprid_i(x)) pari_err_TYPE("idealhnf [fake prime ideal]",x);
157 483942 : return pr_hnf(nf,x); /* PRIME */
158 : }
159 1259805 : switch(tx)
160 : {
161 101922 : case t_MAT:
162 : {
163 : GEN cx;
164 101922 : long nx = lx-1;
165 101922 : N = nf_get_degree(nf);
166 101922 : if (nx == 0) return cgetg(1, t_MAT);
167 101901 : if (nbrows(x) != N) pari_err_TYPE("idealhnf [wrong dimension]",x);
168 101894 : if (nx == 1) return idealhnf_principal(nf, gel(x,1));
169 :
170 85585 : if (nx == N && RgM_is_ZM(x) && ZM_ishnf(x)) return x;
171 49637 : x = Q_primitive_part(x, &cx);
172 49636 : if (nx < N) x = vec_mulid(nf, x, nx, N);
173 49636 : x = ZM_hnfmod(x, ZM_detmult(x));
174 49637 : return cx? ZM_Q_mul(x,cx): x;
175 : }
176 14 : case t_QFB:
177 : {
178 14 : pari_sp av = avma;
179 14 : GEN u, D = nf_get_disc(nf), T = nf_get_pol(nf), f = nf_get_index(nf);
180 14 : GEN A = gel(x,1), B = gel(x,2);
181 14 : N = nf_get_degree(nf);
182 14 : if (N != 2)
183 0 : pari_err_TYPE("idealhnf [Qfb for nonquadratic fields]", x);
184 14 : if (!equalii(qfb_disc(x), D))
185 7 : pari_err_DOMAIN("idealhnf [Qfb]", "disc(q)", "!=", D, x);
186 : /* x -> A Z + (-B + sqrt(D)) / 2 Z
187 : K = Q[t]/T(t), t^2 + ut + v = 0, u^2 - 4v = Df^2
188 : => t = (-u + sqrt(D) f)/2
189 : => sqrt(D)/2 = (t + u/2)/f */
190 7 : u = gel(T,3);
191 7 : B = deg1pol_shallow(ginv(f),
192 : gsub(gdiv(u, shifti(f,1)), gdiv(B,gen_2)),
193 7 : varn(T));
194 7 : return gerepileupto(av, idealhnf_two(nf, mkvec2(A,B)));
195 : }
196 1157869 : default: return idealhnf_principal(nf, x); /* PRINCIPAL */
197 : }
198 : }
199 : /* true nf */
200 : GEN
201 308 : idealhnf(GEN nf, GEN x)
202 : {
203 308 : pari_sp av = avma;
204 308 : GEN y = idealhnf_shallow(nf, x);
205 294 : return (avma == av)? gcopy(y): gerepileupto(av, y);
206 : }
207 :
208 : static GEN
209 84 : nfV_eltembed(GEN nf, GEN x, long prec)
210 518 : { pari_APPLY_type(t_VEC, nfeltembed(nf, gel(x,i), NULL, prec)) }
211 :
212 : static GEN
213 84 : nfweilheight_i(GEN nf, GEN v, long prec)
214 : {
215 84 : long i, l = lg(v);
216 84 : long r1 = nf_get_r1(nf), r2 = nf_get_r2(nf), r12 = r1+r2;
217 84 : GEN id = nfV_idealhnf(nf, v);
218 84 : GEN V = gabs(nfV_eltembed(nf, v, prec), prec);
219 84 : GEN h = gen_1;
220 518 : for (i = 1; i <= r12; i++)
221 : {
222 434 : long j, j0 = 1;
223 2366 : for (j = 2; j < l; j++)
224 1932 : if (cmprr(gmael(V,j,i),gmael(V,j0,i)) > 0) j0 = j;
225 434 : h = gmul(h, i<=r1 ? gmael(V,j0,i): gsqr(gmael(V,j0,i)));
226 : }
227 84 : return gdivgs(glog(gdiv(h, idealnorm(nf, id)), prec), r12+r2);
228 : }
229 :
230 : GEN
231 84 : nfweilheight(GEN nf, GEN v, long prec)
232 : {
233 84 : pari_sp av = avma;
234 84 : checknf(nf);
235 84 : if (!is_vec_t(typ(v)) || lg(v)<2)
236 0 : pari_err_TYPE("nfweilheight",v);
237 84 : return gerepileupto(av, nfweilheight_i(nf, v, prec));
238 : }
239 :
240 : /* GP functions */
241 :
242 : GEN
243 70 : idealtwoelt0(GEN nf, GEN x, GEN a)
244 : {
245 70 : if (!a) return idealtwoelt(nf,x);
246 42 : return idealtwoelt2(nf,x,a);
247 : }
248 :
249 : GEN
250 84 : idealpow0(GEN nf, GEN x, GEN n, long flag)
251 : {
252 84 : if (flag) return idealpowred(nf,x,n);
253 77 : return idealpow(nf,x,n);
254 : }
255 :
256 : GEN
257 70 : idealmul0(GEN nf, GEN x, GEN y, long flag)
258 : {
259 70 : if (flag) return idealmulred(nf,x,y);
260 63 : return idealmul(nf,x,y);
261 : }
262 :
263 : GEN
264 56 : idealdiv0(GEN nf, GEN x, GEN y, long flag)
265 : {
266 56 : switch(flag)
267 : {
268 28 : case 0: return idealdiv(nf,x,y);
269 28 : case 1: return idealdivexact(nf,x,y);
270 0 : default: pari_err_FLAG("idealdiv");
271 : }
272 : return NULL; /* LCOV_EXCL_LINE */
273 : }
274 :
275 : GEN
276 70 : idealaddtoone0(GEN nf, GEN arg1, GEN arg2)
277 : {
278 70 : if (!arg2) return idealaddmultoone(nf,arg1);
279 35 : return idealaddtoone(nf,arg1,arg2);
280 : }
281 :
282 : /* b not a scalar */
283 : static GEN
284 77 : hnf_Z_ZC(GEN nf, GEN a, GEN b) { return hnfmodid(zk_multable(nf,b), a); }
285 : /* b not a scalar */
286 : static GEN
287 70 : hnf_Z_QC(GEN nf, GEN a, GEN b)
288 : {
289 : GEN db;
290 70 : b = Q_remove_denom(b, &db);
291 70 : if (db) a = mulii(a, db);
292 70 : b = hnf_Z_ZC(nf,a,b);
293 70 : return db? RgM_Rg_div(b, db): b;
294 : }
295 : /* b not a scalar (not point in trying to optimize for this case) */
296 : static GEN
297 77 : hnf_Q_QC(GEN nf, GEN a, GEN b)
298 : {
299 : GEN da, db;
300 77 : if (typ(a) == t_INT) return hnf_Z_QC(nf, a, b);
301 7 : da = gel(a,2);
302 7 : a = gel(a,1);
303 7 : b = Q_remove_denom(b, &db);
304 : /* write da = d*A, db = d*B, gcd(A,B) = 1
305 : * gcd(a/(d A), b/(d B)) = gcd(a B, A b) / A B d = gcd(a B, b) / A B d */
306 7 : if (db)
307 : {
308 7 : GEN d = gcdii(da,db);
309 7 : if (!is_pm1(d)) db = diviiexact(db,d); /* B */
310 7 : if (!is_pm1(db))
311 : {
312 7 : a = mulii(a, db); /* a B */
313 7 : da = mulii(da, db); /* A B d = lcm(denom(a),denom(b)) */
314 : }
315 : }
316 7 : return RgM_Rg_div(hnf_Z_ZC(nf,a,b), da);
317 : }
318 : static GEN
319 7 : hnf_QC_QC(GEN nf, GEN a, GEN b)
320 : {
321 : GEN da, db, d, x;
322 7 : a = Q_remove_denom(a, &da);
323 7 : b = Q_remove_denom(b, &db);
324 7 : if (da) b = ZC_Z_mul(b, da);
325 7 : if (db) a = ZC_Z_mul(a, db);
326 7 : d = mul_denom(da, db);
327 7 : a = zk_multable(nf,a); da = zkmultable_capZ(a);
328 7 : b = zk_multable(nf,b); db = zkmultable_capZ(b);
329 7 : x = ZM_hnfmodid(shallowconcat(a,b), gcdii(da,db));
330 7 : return d? RgM_Rg_div(x, d): x;
331 : }
332 : static GEN
333 21 : hnf_Q_Q(GEN nf, GEN a, GEN b) {return scalarmat(Q_gcd(a,b), nf_get_degree(nf));}
334 : GEN
335 217 : idealhnf0(GEN nf, GEN a, GEN b)
336 : {
337 : long ta, tb;
338 : pari_sp av;
339 : GEN x;
340 217 : nf = checknf(nf);
341 217 : if (!b) return idealhnf(nf,a);
342 :
343 : /* HNF of aZ_K+bZ_K */
344 112 : av = avma;
345 112 : a = nf_to_scalar_or_basis(nf,a); ta = typ(a);
346 112 : b = nf_to_scalar_or_basis(nf,b); tb = typ(b);
347 105 : if (ta == t_COL)
348 14 : x = (tb==t_COL)? hnf_QC_QC(nf, a,b): hnf_Q_QC(nf, b,a);
349 : else
350 91 : x = (tb==t_COL)? hnf_Q_QC(nf, a,b): hnf_Q_Q(nf, a,b);
351 105 : return gerepileupto(av, x);
352 : }
353 :
354 : /*******************************************************************/
355 : /* */
356 : /* TWO-ELEMENT FORM */
357 : /* */
358 : /*******************************************************************/
359 : static GEN idealapprfact_i(GEN nf, GEN x, int nored);
360 :
361 : static int
362 229545 : ok_elt(GEN x, GEN xZ, GEN y)
363 : {
364 229545 : pari_sp av = avma;
365 229545 : return gc_bool(av, ZM_equal(x, ZM_hnfmodid(y, xZ)));
366 : }
367 :
368 : /* a + s * b, a and b ZM, s integer */
369 : static GEN
370 67495 : addmul_mat(GEN a, GEN s, GEN b)
371 : {
372 67495 : if (!signe(s)) return a;
373 58648 : if (!equali1(s)) b = ZM_Z_mul(b, s);
374 58648 : return a? ZM_add(a, b): b;
375 : }
376 :
377 : static GEN
378 121847 : get_random_a(GEN nf, GEN x, GEN xZ)
379 : {
380 : pari_sp av;
381 121847 : long i, lm, l = lg(x);
382 : GEN z, beta, mul;
383 :
384 121847 : beta= cgetg(l, t_MAT);
385 121849 : mul = cgetg(l, t_VEC); lm = 1; /* = lg(mul) */
386 : /* look for a in x such that a O/xZ = x O/xZ */
387 255695 : for (i = 2; i < l; i++)
388 : {
389 245732 : GEN xi = gel(x,i);
390 245732 : GEN t = FpM_red(zk_multable(nf,xi), xZ); /* ZM, cannot be a scalar */
391 245728 : if (gequal0(t)) continue;
392 200703 : if (ok_elt(x,xZ, t)) return xi;
393 88818 : gel(beta,lm) = xi;
394 : /* mul[i] = { canonical generators for x[i] O/xZ as Z-module } */
395 88818 : gel(mul,lm) = t; lm++;
396 : }
397 9963 : setlg(mul, lm);
398 9963 : setlg(beta,lm); z = cgetg(lm, t_VEC);
399 30678 : for(av = avma;; set_avma(av))
400 20715 : {
401 30678 : GEN a = NULL;
402 98173 : for (i = 1; i < lm; i++)
403 : {
404 67495 : gel(z,i) = randomi(xZ);
405 67495 : a = addmul_mat(a, gel(z,i), gel(mul,i));
406 : }
407 : /* a = matrix (NOT HNF) of ideal generated by beta.z in O/xZ */
408 30678 : if (a && ok_elt(x,xZ, a)) break;
409 : }
410 9963 : return ZM_ZC_mul(beta, z);
411 : }
412 :
413 : /* x square matrix, assume it is HNF */
414 : static GEN
415 257754 : mat_ideal_two_elt(GEN nf, GEN x)
416 : {
417 : GEN y, a, cx, xZ;
418 257754 : long N = nf_get_degree(nf);
419 : pari_sp av, tetpil;
420 :
421 257754 : if (lg(x)-1 != N) pari_err_DIM("idealtwoelt");
422 257740 : if (N == 2) return mkvec2copy(gcoeff(x,1,1), gel(x,2));
423 :
424 138035 : y = cgetg(3,t_VEC); av = avma;
425 138035 : cx = Q_content(x);
426 138033 : xZ = gcoeff(x,1,1);
427 138033 : if (gequal(xZ, cx)) /* x = (cx) */
428 : {
429 5341 : gel(y,1) = cx;
430 5341 : gel(y,2) = gen_0; return y;
431 : }
432 132692 : if (equali1(cx)) cx = NULL;
433 : else
434 : {
435 3744 : x = Q_div_to_int(x, cx);
436 3744 : xZ = gcoeff(x,1,1);
437 : }
438 132692 : if (N < 6)
439 113439 : a = get_random_a(nf, x, xZ);
440 : else
441 : {
442 19253 : const long FB[] = { _evallg(15+1) | evaltyp(t_VECSMALL),
443 : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
444 : };
445 19253 : GEN P, E, a1 = Z_lsmoothen(xZ, (GEN)FB, &P, &E);
446 19253 : if (!a1) /* factors completely */
447 10845 : a = idealapprfact_i(nf, idealfactor(nf,x), 1);
448 8408 : else if (lg(P) == 1) /* no small factors */
449 2603 : a = get_random_a(nf, x, xZ);
450 : else /* general case */
451 : {
452 : GEN A0, A1, a0, u0, u1, v0, v1, pi0, pi1, t, u;
453 5805 : a0 = diviiexact(xZ, a1);
454 5805 : A0 = ZM_hnfmodid(x, a0); /* smooth part of x */
455 5805 : A1 = ZM_hnfmodid(x, a1); /* cofactor */
456 5805 : pi0 = idealapprfact_i(nf, idealfactor(nf,A0), 1);
457 5805 : pi1 = get_random_a(nf, A1, a1);
458 5805 : (void)bezout(a0, a1, &v0,&v1);
459 5805 : u0 = mulii(a0, v0);
460 5805 : u1 = mulii(a1, v1);
461 5805 : if (typ(pi0) != t_COL) t = addmulii(u0, pi0, u1);
462 : else
463 5805 : { t = ZC_Z_mul(pi0, u1); gel(t,1) = addii(gel(t,1), u0); }
464 5805 : u = ZC_Z_mul(pi1, u0); gel(u,1) = addii(gel(u,1), u1);
465 5805 : a = nfmuli(nf, centermod(u, xZ), centermod(t, xZ));
466 : }
467 : }
468 132695 : if (cx)
469 : {
470 3744 : a = centermod(a, xZ);
471 3744 : tetpil = avma;
472 3744 : if (typ(cx) == t_INT)
473 : {
474 98 : gel(y,1) = mulii(xZ, cx);
475 98 : gel(y,2) = ZC_Z_mul(a, cx);
476 : }
477 : else
478 : {
479 3646 : gel(y,1) = gmul(xZ, cx);
480 3646 : gel(y,2) = RgC_Rg_mul(a, cx);
481 : }
482 : }
483 : else
484 : {
485 128951 : tetpil = avma;
486 128951 : gel(y,1) = icopy(xZ);
487 128950 : gel(y,2) = centermod(a, xZ);
488 : }
489 132693 : gerepilecoeffssp(av,tetpil,y+1,2); return y;
490 : }
491 :
492 : /* Given an ideal x, returns [a,alpha] such that a is in Q,
493 : * x = a Z_K + alpha Z_K, alpha in K^*
494 : * a = 0 or alpha = 0 are possible, but do not try to determine whether
495 : * x is principal. */
496 : GEN
497 102866 : idealtwoelt(GEN nf, GEN x)
498 : {
499 : pari_sp av;
500 102866 : long tx = idealtyp(&x, NULL);
501 102859 : nf = checknf(nf);
502 102859 : if (tx == id_MAT) return mat_ideal_two_elt(nf,x);
503 1015 : if (tx == id_PRIME) return mkvec2copy(gel(x,1), gel(x,2));
504 : /* id_PRINCIPAL */
505 1008 : av = avma; x = nf_to_scalar_or_basis(nf, x);
506 1820 : return gerepilecopy(av, typ(x)==t_COL? mkvec2(gen_0,x):
507 903 : mkvec2(Q_abs_shallow(x),gen_0));
508 : }
509 :
510 : /*******************************************************************/
511 : /* */
512 : /* FACTORIZATION */
513 : /* */
514 : /*******************************************************************/
515 : /* x integral ideal in HNF, Zval = v_p(x \cap Z) > 0; return v_p(Nx) */
516 : static long
517 3990828 : idealHNF_norm_pval(GEN x, GEN p, long Zval)
518 : {
519 3990828 : long i, v = Zval, l = lg(x);
520 31631240 : for (i = 2; i < l; i++) v += Z_pval(gcoeff(x,i,i), p);
521 3990825 : return v;
522 : }
523 :
524 : /* x integral in HNF, f0 = partial factorization of a multiple of
525 : * x[1,1] = x\cap Z */
526 : GEN
527 284532 : idealHNF_Z_factor_i(GEN x, GEN f0, GEN *pvN, GEN *pvZ)
528 : {
529 284532 : GEN P, E, vN, vZ, xZ = gcoeff(x,1,1), f = f0? f0: Z_factor(xZ);
530 : long i, l;
531 284541 : P = gel(f,1); l = lg(P);
532 284541 : E = gel(f,2);
533 284541 : *pvN = vN = cgetg(l, t_VECSMALL);
534 284552 : *pvZ = vZ = cgetg(l, t_VECSMALL);
535 708535 : for (i = 1; i < l; i++)
536 : {
537 423986 : GEN p = gel(P,i);
538 423986 : vZ[i] = f0? Z_pval(xZ, p): (long) itou(gel(E,i));
539 423985 : vN[i] = idealHNF_norm_pval(x,p, vZ[i]);
540 : }
541 284549 : return P;
542 : }
543 : /* return P, primes dividing Nx and xZ = x\cap Z, set v_p(Nx), v_p(xZ);
544 : * x integral in HNF */
545 : GEN
546 0 : idealHNF_Z_factor(GEN x, GEN *pvN, GEN *pvZ)
547 0 : { return idealHNF_Z_factor_i(x, NULL, pvN, pvZ); }
548 :
549 : /* v_P(A)*f(P) <= Nval [e.g. Nval = v_p(Norm A)], Zval = v_p(A \cap Z).
550 : * Return v_P(A) */
551 : static long
552 4127046 : idealHNF_val(GEN A, GEN P, long Nval, long Zval)
553 : {
554 4127046 : long f = pr_get_f(P), vmax, v, e, i, j, k, l;
555 : GEN mul, B, a, y, r, p, pk, cx, vals;
556 : pari_sp av;
557 :
558 4127044 : if (Nval < f) return 0;
559 4123867 : p = pr_get_p(P);
560 4123866 : e = pr_get_e(P);
561 : /* v_P(A) <= max [ e * v_p(A \cap Z), floor[v_p(Nix) / f ] */
562 4123871 : vmax = minss(Zval * e, Nval / f);
563 4123872 : mul = pr_get_tau(P);
564 4123866 : l = lg(mul);
565 4123866 : B = cgetg(l,t_MAT);
566 : /* B[1] not needed: v_pr(A[1]) = v_pr(A \cap Z) is known already */
567 4130127 : gel(B,1) = gen_0; /* dummy */
568 23187632 : for (j = 2; j < l; j++)
569 : {
570 20862817 : GEN x = gel(A,j);
571 20862817 : gel(B,j) = y = cgetg(l, t_COL);
572 227672322 : for (i = 1; i < l; i++)
573 : { /* compute a = (x.t0)_i, A in HNF ==> x[j+1..l-1] = 0 */
574 208614817 : a = mulii(gel(x,1), gcoeff(mul,i,1));
575 1461126848 : for (k = 2; k <= j; k++) a = addii(a, mulii(gel(x,k), gcoeff(mul,i,k)));
576 : /* p | a ? */
577 208604981 : gel(y,i) = dvmdii(a,p,&r); if (signe(r)) return 0;
578 : }
579 : }
580 2324815 : vals = cgetg(l, t_VECSMALL);
581 : /* vals[1] not needed */
582 16538073 : for (j = 2; j < l; j++)
583 : {
584 14213102 : gel(B,j) = Q_primitive_part(gel(B,j), &cx);
585 14213150 : vals[j] = cx? 1 + e * Q_pval(cx, p): 1;
586 : }
587 2324971 : pk = powiu(p, ceildivuu(vmax, e));
588 2324881 : av = avma; y = cgetg(l,t_COL);
589 : /* can compute mod p^ceil((vmax-v)/e) */
590 3446032 : for (v = 1; v < vmax; v++)
591 : { /* we know v_pr(Bj) >= v for all j */
592 1151132 : if (e == 1 || (vmax - v) % e == 0) pk = diviiexact(pk, p);
593 7442217 : for (j = 2; j < l; j++)
594 : {
595 6321074 : GEN x = gel(B,j); if (v < vals[j]) continue;
596 43381445 : for (i = 1; i < l; i++)
597 : {
598 39287615 : pari_sp av2 = avma;
599 39287615 : a = mulii(gel(x,1), gcoeff(mul,i,1));
600 517386474 : for (k = 2; k < l; k++) a = addii(a, mulii(gel(x,k), gcoeff(mul,i,k)));
601 : /* a = (x.t_0)_i; p | a ? */
602 39287359 : a = dvmdii(a,p,&r); if (signe(r)) return v;
603 39257435 : if (lgefint(a) > lgefint(pk)) a = remii(a, pk);
604 39257435 : gel(y,i) = gerepileuptoint(av2, a);
605 : }
606 4093830 : gel(B,j) = y; y = x;
607 4093830 : if (gc_needed(av,3))
608 : {
609 0 : if(DEBUGMEM>1) pari_warn(warnmem,"idealval");
610 0 : gerepileall(av,3, &y,&B,&pk);
611 : }
612 : }
613 : }
614 2294900 : return v;
615 : }
616 : /* true nf, x != 0 integral ideal in HNF, cx t_INT or NULL,
617 : * FA integer factorization matrix or NULL. Return partial factorization of
618 : * cx * x above primes in FA (complete factorization if !FA)*/
619 : static GEN
620 284532 : idealHNF_factor_i(GEN nf, GEN x, GEN cx, GEN FA)
621 : {
622 284532 : const long N = lg(x)-1;
623 : long i, j, k, l, v;
624 284532 : GEN vN, vZ, vP, vE, vp = idealHNF_Z_factor_i(x, FA, &vN,&vZ);
625 :
626 284549 : l = lg(vp);
627 284549 : i = cx? expi(cx)+1: 1;
628 284555 : vP = cgetg((l+i-2)*N+1, t_COL);
629 284546 : vE = cgetg((l+i-2)*N+1, t_COL);
630 708532 : for (i = k = 1; i < l; i++)
631 : {
632 423986 : GEN L, p = gel(vp,i);
633 423986 : long Nval = vN[i], Zval = vZ[i], vc = cx? Z_pvalrem(cx,p,&cx): 0;
634 423984 : if (vc)
635 : {
636 43254 : L = idealprimedec(nf,p);
637 43254 : if (is_pm1(cx)) cx = NULL;
638 : }
639 : else
640 380730 : L = idealprimedec_limit_f(nf,p,Nval);
641 984171 : for (j = 1; Nval && j < lg(L); j++) /* !Nval => only cx contributes */
642 : {
643 560190 : GEN P = gel(L,j);
644 560190 : pari_sp av = avma;
645 560190 : v = idealHNF_val(x, P, Nval, Zval);
646 560178 : set_avma(av);
647 560182 : Nval -= v*pr_get_f(P);
648 560184 : v += vc * pr_get_e(P); if (!v) continue;
649 471245 : gel(vP,k) = P;
650 471245 : gel(vE,k) = utoipos(v); k++;
651 : }
652 466681 : if (vc) for (; j<lg(L); j++)
653 : {
654 42707 : GEN P = gel(L,j);
655 42707 : gel(vP,k) = P;
656 42707 : gel(vE,k) = utoipos(vc * pr_get_e(P)); k++;
657 : }
658 : }
659 284546 : if (cx && !FA)
660 : { /* complete factorization */
661 73505 : GEN f = Z_factor(cx), cP = gel(f,1), cE = gel(f,2);
662 73506 : long lc = lg(cP);
663 159857 : for (i=1; i<lc; i++)
664 : {
665 86351 : GEN p = gel(cP,i), L = idealprimedec(nf,p);
666 86351 : long vc = itos(gel(cE,i));
667 189089 : for (j=1; j<lg(L); j++)
668 : {
669 102738 : GEN P = gel(L,j);
670 102738 : gel(vP,k) = P;
671 102738 : gel(vE,k) = utoipos(vc * pr_get_e(P)); k++;
672 : }
673 : }
674 : }
675 284547 : setlg(vP, k);
676 284544 : setlg(vE, k); return mkmat2(vP, vE);
677 : }
678 : /* true nf, x integral ideal */
679 : static GEN
680 239089 : idealHNF_factor(GEN nf, GEN x, ulong lim)
681 : {
682 239089 : GEN cx, F = NULL;
683 239089 : if (lim)
684 : {
685 : GEN P, E;
686 : long i;
687 : /* strict useless because of prime table */
688 70 : F = absZ_factor_limit(gcoeff(x,1,1), lim);
689 70 : P = gel(F,1);
690 70 : E = gel(F,2);
691 : /* filter out entries > lim */
692 119 : for (i = lg(P)-1; i; i--)
693 119 : if (cmpiu(gel(P,i), lim) <= 0) break;
694 70 : setlg(P, i+1);
695 70 : setlg(E, i+1);
696 : }
697 239089 : x = Q_primitive_part(x, &cx);
698 239071 : return idealHNF_factor_i(nf, x, cx, F);
699 : }
700 : /* c * vector(#L,i,L[i].e), assume results fit in ulong */
701 : static GEN
702 27520 : prV_e_muls(GEN L, long c)
703 : {
704 27520 : long j, l = lg(L);
705 27520 : GEN z = cgetg(l, t_COL);
706 56621 : for (j = 1; j < l; j++) gel(z,j) = stoi(c * pr_get_e(gel(L,j)));
707 27501 : return z;
708 : }
709 : /* true nf, y in Q */
710 : static GEN
711 26903 : Q_nffactor(GEN nf, GEN y, ulong lim)
712 : {
713 : GEN f, P, E;
714 : long l, i;
715 26903 : if (typ(y) == t_INT)
716 : {
717 26875 : if (!signe(y)) pari_err_DOMAIN("idealfactor", "ideal", "=",gen_0,y);
718 26861 : if (is_pm1(y)) return trivial_fact();
719 : }
720 17174 : y = Q_abs_shallow(y);
721 17174 : if (!lim) f = Q_factor(y);
722 : else
723 : {
724 154 : f = Q_factor_limit(y, lim);
725 154 : P = gel(f,1);
726 154 : E = gel(f,2);
727 224 : for (i = lg(P)-1; i > 0; i--)
728 203 : if (abscmpiu(gel(P,i), lim) < 0) break;
729 154 : setlg(P,i+1); setlg(E,i+1);
730 : }
731 17165 : P = gel(f,1); l = lg(P); if (l == 1) return f;
732 17144 : E = gel(f,2);
733 44684 : for (i = 1; i < l; i++)
734 : {
735 27556 : gel(P,i) = idealprimedec(nf, gel(P,i));
736 27520 : gel(E,i) = prV_e_muls(gel(P,i), itos(gel(E,i)));
737 : }
738 17128 : P = shallowconcat1(P); gel(f,1) = P; settyp(P, t_COL);
739 17134 : E = shallowconcat1(E); gel(f,2) = E; return f;
740 : }
741 :
742 : GEN
743 25424 : idealfactor_partial(GEN nf, GEN x, GEN L)
744 : {
745 25424 : pari_sp av = avma;
746 : long i, j, l;
747 : GEN P, E;
748 25424 : if (!L) return idealfactor(nf, x);
749 24584 : if (typ(L) == t_INT) return idealfactor_limit(nf, x, itou(L));
750 24556 : l = lg(L); if (l == 1) return trivial_fact();
751 23772 : P = cgetg(l, t_VEC);
752 89467 : for (i = 1; i < l; i++)
753 : {
754 65695 : GEN p = gel(L,i);
755 65695 : gel(P,i) = typ(p) == t_INT? idealprimedec(nf, p): mkvec(p);
756 : }
757 23772 : P = shallowconcat1(P); settyp(P, t_COL);
758 23772 : P = gen_sort_uniq(P, (void*)&cmp_prime_ideal, &cmp_nodata);
759 23772 : E = cgetg_copy(P, &l);
760 113974 : for (i = j = 1; i < l; i++)
761 : {
762 90202 : long v = idealval(nf, x, gel(P,i));
763 90202 : if (v) { gel(P,j) = gel(P,i); gel(E,j) = stoi(v); j++; }
764 : }
765 23772 : setlg(P,j);
766 23772 : setlg(E,j); return gerepilecopy(av, mkmat2(P, E));
767 : }
768 : GEN
769 266094 : idealfactor_limit(GEN nf, GEN x, ulong lim)
770 : {
771 266094 : pari_sp av = avma;
772 : GEN fa, y;
773 266094 : long tx = idealtyp(&x, NULL);
774 :
775 266090 : if (tx == id_PRIME)
776 : {
777 119 : if (lim && abscmpiu(pr_get_p(x), lim) >= 0) return trivial_fact();
778 112 : retmkmat2(mkcolcopy(x), mkcol(gen_1));
779 : }
780 265971 : nf = checknf(nf);
781 265972 : if (tx == id_PRINCIPAL)
782 : {
783 28156 : y = nf_to_scalar_or_basis(nf, x);
784 28156 : if (typ(y) != t_COL) return gerepilecopy(av, Q_nffactor(nf, y, lim));
785 : }
786 239069 : y = idealnumden(nf, x);
787 239075 : fa = idealHNF_factor(nf, gel(y,1), lim);
788 239066 : if (!isint1(gel(y,2)))
789 14 : fa = famat_div_shallow(fa, idealHNF_factor(nf, gel(y,2), lim));
790 239066 : fa = gerepilecopy(av, fa);
791 239074 : return sort_factor(fa, (void*)&cmp_prime_ideal, &cmp_nodata);
792 : }
793 : GEN
794 265716 : idealfactor(GEN nf, GEN x) { return idealfactor_limit(nf, x, 0); }
795 : GEN
796 182 : gpidealfactor(GEN nf, GEN x, GEN lim)
797 : {
798 182 : ulong L = 0;
799 182 : if (lim)
800 : {
801 70 : if (typ(lim) != t_INT || signe(lim) < 0) pari_err_FLAG("idealfactor");
802 70 : L = itou(lim);
803 : }
804 182 : return idealfactor_limit(nf, x, L);
805 : }
806 :
807 : static GEN
808 7257 : ramified_root(GEN nf, GEN R, GEN A, long n)
809 : {
810 7257 : GEN v, P = gel(idealfactor(nf, R), 1);
811 7257 : long i, l = lg(P);
812 7257 : v = cgetg(l, t_VECSMALL);
813 7901 : for (i = 1; i < l; i++)
814 : {
815 651 : long w = idealval(nf, A, gel(P,i));
816 651 : if (w % n) return NULL;
817 644 : v[i] = w / n;
818 : }
819 7250 : return idealfactorback(nf, P, v, 0);
820 : }
821 : static int
822 7 : ramified_root_simple(GEN nf, long n, GEN P, GEN v)
823 : {
824 7 : long i, l = lg(v);
825 21 : for (i = 1; i < l; i++)
826 : {
827 14 : long w = v[i] % n;
828 14 : if (w)
829 : {
830 7 : GEN vpr = idealprimedec(nf, gel(P,i));
831 7 : long lpr = lg(vpr), j;
832 14 : for (j = 1; j < lpr; j++)
833 : {
834 7 : long e = pr_get_e(gel(vpr,j));
835 7 : if ((e * w) % n) return 0;
836 : }
837 : }
838 : }
839 7 : return 1;
840 : }
841 : /* true nf, n > 1, A a non-zero integral ideal; check whether A is the n-th
842 : * power of an ideal and set *pB to its n-th root if so */
843 : static long
844 7264 : idealsqrtn_int(GEN nf, GEN A, long n, GEN *pB)
845 : {
846 : GEN C, root;
847 : long i, l;
848 :
849 7264 : if (typ(A) == t_MAT && ZM_isscalar(A, NULL)) A = gcoeff(A,1,1);
850 7264 : if (typ(A) == t_INT) /* > 0 */
851 : {
852 5060 : GEN P = nf_get_ramified_primes(nf), v, q;
853 5060 : l = lg(P); v = cgetg(l, t_VECSMALL);
854 23966 : for (i = 1; i < l; i++) v[i] = Z_pvalrem(A, gel(P,i), &A);
855 5060 : C = gen_1;
856 5060 : if (!isint1(A) && !Z_ispowerall(A, n, pB? &C: NULL)) return 0;
857 5060 : if (!pB) return ramified_root_simple(nf, n, P, v);
858 5053 : q = factorback2(P, v);
859 5053 : root = ramified_root(nf, q, q, n);
860 5053 : if (!root) return 0;
861 5053 : if (!equali1(C)) root = isint1(root)? C: ZM_Z_mul(root, C);
862 5053 : *pB = root; return 1;
863 : }
864 : /* compute valuations at ramified primes */
865 2204 : root = ramified_root(nf, idealadd(nf, nf_get_diff(nf), A), A, n);
866 2204 : if (!root) return 0;
867 : /* remove ramified primes */
868 2197 : if (isint1(root))
869 1854 : root = matid(nf_get_degree(nf));
870 : else
871 343 : A = idealdivexact(nf, A, idealpows(nf,root,n));
872 2197 : A = Q_primitive_part(A, &C);
873 2197 : if (C)
874 : {
875 7 : if (!Z_ispowerall(C,n,&C)) return 0;
876 0 : if (pB) root = ZM_Z_mul(root, C);
877 : }
878 :
879 : /* compute final n-th root, at most degree(nf)-1 iterations */
880 2190 : for (i = 0;; i++)
881 2064 : {
882 4254 : GEN J, b, a = gcoeff(A,1,1); /* A \cap Z */
883 4254 : if (is_pm1(a)) break;
884 2092 : if (!Z_ispowerall(a,n,&b)) return 0;
885 2064 : J = idealadd(nf, b, A);
886 2064 : A = idealdivexact(nf, idealpows(nf,J,n), A);
887 : /* div and not divexact here */
888 2064 : if (pB) root = odd(i)? idealdiv(nf, root, J): idealmul(nf, root, J);
889 : }
890 2162 : if (pB) *pB = root;
891 2162 : return 1;
892 : }
893 :
894 : /* A is assumed to be the n-th power of an ideal in nf
895 : returns its n-th root. */
896 : long
897 3660 : idealispower(GEN nf, GEN A, long n, GEN *pB)
898 : {
899 3660 : pari_sp av = avma;
900 : GEN v, N, D;
901 3660 : nf = checknf(nf);
902 3660 : if (n <= 0) pari_err_DOMAIN("idealispower", "n", "<=", gen_0, stoi(n));
903 3660 : if (n == 1) { if (pB) *pB = idealhnf(nf,A); return 1; }
904 3653 : v = idealnumden(nf,A);
905 3653 : if (gequal0(gel(v,1))) { set_avma(av); if (pB) *pB = cgetg(1,t_MAT); return 1; }
906 3653 : if (!idealsqrtn_int(nf, gel(v,1), n, pB? &N: NULL)) return 0;
907 3611 : if (!idealsqrtn_int(nf, gel(v,2), n, pB? &D: NULL)) return 0;
908 3611 : if (pB) *pB = gerepileupto(av, idealdiv(nf,N,D)); else set_avma(av);
909 3611 : return 1;
910 : }
911 :
912 : /* x t_INT or integral nonzero ideal in HNF */
913 : static GEN
914 97230 : idealredmodpower_i(GEN nf, GEN x, ulong k, ulong B)
915 : {
916 : GEN cx, y, U, N, F, Q;
917 97230 : if (typ(x) == t_INT)
918 : {
919 51212 : if (!signe(x) || is_pm1(x)) return gen_1;
920 1918 : F = Z_factor_limit(x, B);
921 1918 : gel(F,2) = gdiventgs(gel(F,2), k);
922 1918 : return ginv(factorback(F));
923 : }
924 46018 : N = gcoeff(x,1,1); if (is_pm1(N)) return gen_1;
925 45471 : F = absZ_factor_limit_strict(N, B, &U);
926 45471 : if (U)
927 : {
928 146 : GEN M = powii(gel(U,1), gel(U,2));
929 146 : y = hnfmodid(x, M); /* coprime part to B! */
930 146 : if (!idealispower(nf, y, k, &U)) U = NULL;
931 146 : x = hnfmodid(x, diviiexact(N, M));
932 : }
933 : /* x = B-smooth part of initial x */
934 45471 : x = Q_primitive_part(x, &cx);
935 45468 : F = idealHNF_factor_i(nf, x, cx, F);
936 45471 : gel(F,2) = gdiventgs(gel(F,2), k);
937 45471 : Q = idealfactorback(nf, gel(F,1), gel(F,2), 0);
938 45471 : if (U) Q = idealmul(nf,Q,U);
939 45471 : if (typ(Q) == t_INT) return Q;
940 10865 : y = idealred_elt(nf, idealHNF_inv_Z(nf, Q));
941 10865 : return gdiv(y, gcoeff(Q,1,1));
942 : }
943 : GEN
944 48621 : idealredmodpower(GEN nf, GEN x, ulong n, ulong B)
945 : {
946 48621 : pari_sp av = avma;
947 : GEN a, b;
948 48621 : nf = checknf(nf);
949 48621 : if (!n) pari_err_DOMAIN("idealredmodpower","n", "=", gen_0, gen_0);
950 48621 : x = idealnumden(nf, x);
951 48622 : a = gel(x,1);
952 48622 : if (isintzero(a)) { set_avma(av); return gen_1; }
953 48615 : a = idealredmodpower_i(nf, gel(x,1), n, B);
954 48615 : b = idealredmodpower_i(nf, gel(x,2), n, B);
955 48614 : if (!isint1(b)) a = nf_to_scalar_or_basis(nf, nfdiv(nf, a, b));
956 48614 : return gerepilecopy(av, a);
957 : }
958 :
959 : /* P prime ideal in idealprimedec format. Return valuation(A) at P */
960 : long
961 9470872 : idealval(GEN nf, GEN A, GEN P)
962 : {
963 9470872 : pari_sp av = avma;
964 : GEN p, cA;
965 9470872 : long vcA, v, Zval, tx = idealtyp(&A, NULL);
966 :
967 9470859 : if (tx == id_PRINCIPAL) return nfval(nf,A,P);
968 9371792 : checkprid(P);
969 9371948 : if (tx == id_PRIME) return pr_equal(P, A)? 1: 0;
970 : /* id_MAT */
971 9371920 : nf = checknf(nf);
972 9372082 : A = Q_primitive_part(A, &cA);
973 9372200 : p = pr_get_p(P);
974 9372218 : vcA = cA? Q_pval(cA,p): 0;
975 9372221 : if (pr_is_inert(P)) return gc_long(av,vcA);
976 8973827 : Zval = Z_pval(gcoeff(A,1,1), p);
977 8973851 : if (!Zval) v = 0;
978 : else
979 : {
980 3566771 : long Nval = idealHNF_norm_pval(A, p, Zval);
981 3566860 : v = idealHNF_val(A, P, Nval, Zval);
982 : }
983 8973779 : return gc_long(av, vcA? v + vcA*pr_get_e(P): v);
984 : }
985 : GEN
986 7119 : gpidealval(GEN nf, GEN ix, GEN P)
987 : {
988 7119 : long v = idealval(nf,ix,P);
989 7105 : return v == LONG_MAX? mkoo(): stoi(v);
990 : }
991 :
992 : /* gcd and generalized Bezout */
993 :
994 : GEN
995 108265 : idealadd(GEN nf, GEN x, GEN y)
996 : {
997 108265 : pari_sp av = avma;
998 : long tx, ty;
999 : GEN z, a, dx, dy, dz;
1000 :
1001 108265 : tx = idealtyp(&x, NULL);
1002 108265 : ty = idealtyp(&y, NULL); nf = checknf(nf);
1003 108265 : if (tx != id_MAT) x = idealhnf_shallow(nf,x);
1004 108265 : if (ty != id_MAT) y = idealhnf_shallow(nf,y);
1005 108265 : if (lg(x) == 1) return gerepilecopy(av,y);
1006 107271 : if (lg(y) == 1) return gerepilecopy(av,x); /* check for 0 ideal */
1007 106816 : dx = Q_denom(x);
1008 106816 : dy = Q_denom(y); dz = lcmii(dx,dy);
1009 106816 : if (is_pm1(dz)) dz = NULL; else {
1010 15834 : x = Q_muli_to_int(x, dz);
1011 15834 : y = Q_muli_to_int(y, dz);
1012 : }
1013 106816 : a = gcdii(gcoeff(x,1,1), gcoeff(y,1,1));
1014 106816 : if (is_pm1(a))
1015 : {
1016 37918 : long N = lg(x)-1;
1017 37918 : if (!dz) { set_avma(av); return matid(N); }
1018 3906 : return gerepileupto(av, scalarmat(ginv(dz), N));
1019 : }
1020 68898 : z = ZM_hnfmodid(shallowconcat(x,y), a);
1021 68898 : if (dz) z = RgM_Rg_div(z,dz);
1022 68898 : return gerepileupto(av,z);
1023 : }
1024 :
1025 : static GEN
1026 28 : trivial_merge(GEN x)
1027 28 : { return (lg(x) == 1 || !is_pm1(gcoeff(x,1,1)))? NULL: gen_1; }
1028 : /* true nf */
1029 : static GEN
1030 731771 : _idealaddtoone(GEN nf, GEN x, GEN y, long red)
1031 : {
1032 : GEN a;
1033 731771 : long tx = idealtyp(&x, NULL);
1034 731757 : long ty = idealtyp(&y, NULL);
1035 : long ea;
1036 731752 : if (tx != id_MAT) x = idealhnf_shallow(nf, x);
1037 731767 : if (ty != id_MAT) y = idealhnf_shallow(nf, y);
1038 731767 : if (lg(x) == 1)
1039 14 : a = trivial_merge(y);
1040 731753 : else if (lg(y) == 1)
1041 14 : a = trivial_merge(x);
1042 : else
1043 731739 : a = hnfmerge_get_1(x, y);
1044 731714 : if (!a) pari_err_COPRIME("idealaddtoone",x,y);
1045 731700 : if (red && (ea = gexpo(a)) > 10)
1046 : {
1047 5152 : GEN b = (typ(a) == t_COL)? a: scalarcol_shallow(a, nf_get_degree(nf));
1048 5152 : b = ZC_reducemodlll(b, idealHNF_mul(nf,x,y));
1049 5152 : if (gexpo(b) < ea) a = b;
1050 : }
1051 731700 : return a;
1052 : }
1053 : /* true nf */
1054 : GEN
1055 19768 : idealaddtoone_i(GEN nf, GEN x, GEN y)
1056 19768 : { return _idealaddtoone(nf, x, y, 1); }
1057 : /* true nf */
1058 : GEN
1059 712003 : idealaddtoone_raw(GEN nf, GEN x, GEN y)
1060 712003 : { return _idealaddtoone(nf, x, y, 0); }
1061 :
1062 : GEN
1063 98 : idealaddtoone(GEN nf, GEN x, GEN y)
1064 : {
1065 98 : GEN z = cgetg(3,t_VEC), a;
1066 98 : pari_sp av = avma;
1067 98 : nf = checknf(nf);
1068 98 : a = gerepileupto(av, idealaddtoone_i(nf,x,y));
1069 84 : gel(z,1) = a;
1070 84 : gel(z,2) = typ(a) == t_COL? Z_ZC_sub(gen_1,a): subui(1,a);
1071 84 : return z;
1072 : }
1073 :
1074 : /* assume elements of list are integral ideals */
1075 : GEN
1076 35 : idealaddmultoone(GEN nf, GEN list)
1077 : {
1078 35 : pari_sp av = avma;
1079 35 : long N, i, l, nz, tx = typ(list);
1080 : GEN H, U, perm, L;
1081 :
1082 35 : nf = checknf(nf); N = nf_get_degree(nf);
1083 35 : if (!is_vec_t(tx)) pari_err_TYPE("idealaddmultoone",list);
1084 35 : l = lg(list);
1085 35 : L = cgetg(l, t_VEC);
1086 35 : if (l == 1)
1087 0 : pari_err_DOMAIN("idealaddmultoone", "sum(ideals)", "!=", gen_1, L);
1088 35 : nz = 0; /* number of nonzero ideals in L */
1089 98 : for (i=1; i<l; i++)
1090 : {
1091 70 : GEN I = gel(list,i);
1092 70 : if (typ(I) != t_MAT) I = idealhnf_shallow(nf,I);
1093 70 : if (lg(I) != 1)
1094 : {
1095 42 : nz++; RgM_check_ZM(I,"idealaddmultoone");
1096 35 : if (lgcols(I) != N+1) pari_err_TYPE("idealaddmultoone [not an ideal]", I);
1097 : }
1098 63 : gel(L,i) = I;
1099 : }
1100 28 : H = ZM_hnfperm(shallowconcat1(L), &U, &perm);
1101 28 : if (lg(H) == 1 || !equali1(gcoeff(H,1,1)))
1102 7 : pari_err_DOMAIN("idealaddmultoone", "sum(ideals)", "!=", gen_1, L);
1103 49 : for (i=1; i<=N; i++)
1104 49 : if (perm[i] == 1) break;
1105 21 : U = gel(U,(nz-1)*N + i); /* (L[1]|...|L[nz]) U = 1 */
1106 21 : nz = 0;
1107 63 : for (i=1; i<l; i++)
1108 : {
1109 42 : GEN c = gel(L,i);
1110 42 : if (lg(c) == 1)
1111 14 : c = gen_0;
1112 : else {
1113 28 : c = ZM_ZC_mul(c, vecslice(U, nz*N + 1, (nz+1)*N));
1114 28 : nz++;
1115 : }
1116 42 : gel(L,i) = c;
1117 : }
1118 21 : return gerepilecopy(av, L);
1119 : }
1120 :
1121 : /* multiplication */
1122 :
1123 : /* x integral ideal (without archimedean component) in HNF form
1124 : * y = [a,alpha] corresponds to the integral ideal aZ_K+alpha Z_K, a in Z,
1125 : * alpha a ZV or a ZM (multiplication table). Multiply them */
1126 : static GEN
1127 799811 : idealHNF_mul_two(GEN nf, GEN x, GEN y)
1128 : {
1129 799811 : GEN m, a = gel(y,1), alpha = gel(y,2);
1130 : long i, N;
1131 :
1132 799811 : if (typ(alpha) != t_MAT)
1133 : {
1134 462057 : alpha = zk_scalar_or_multable(nf, alpha);
1135 462047 : if (typ(alpha) == t_INT) /* e.g. y inert ? 0 should not (but may) occur */
1136 14048 : return signe(a)? ZM_Z_mul(x, gcdii(a, alpha)): cgetg(1,t_MAT);
1137 : }
1138 785753 : N = lg(x)-1; m = cgetg((N<<1)+1,t_MAT);
1139 3272379 : for (i=1; i<=N; i++) gel(m,i) = ZM_ZC_mul(alpha,gel(x,i));
1140 3271731 : for (i=1; i<=N; i++) gel(m,i+N) = ZC_Z_mul(gel(x,i), a);
1141 785283 : return ZM_hnfmodid(m, mulii(a, gcoeff(x,1,1)));
1142 : }
1143 :
1144 : /* Assume x and y are integral in HNF form [NOT extended]. Not memory clean.
1145 : * HACK: ideal in y can be of the form [a,b], a in Z, b in Z_K */
1146 : GEN
1147 388693 : idealHNF_mul(GEN nf, GEN x, GEN y)
1148 : {
1149 : GEN z;
1150 388693 : if (typ(y) == t_VEC)
1151 208761 : z = idealHNF_mul_two(nf,x,y);
1152 : else
1153 : { /* reduce one ideal to two-elt form. The smallest */
1154 179932 : GEN xZ = gcoeff(x,1,1), yZ = gcoeff(y,1,1);
1155 179932 : if (cmpii(xZ, yZ) < 0)
1156 : {
1157 39146 : if (is_pm1(xZ)) return gcopy(y);
1158 23070 : z = idealHNF_mul_two(nf, y, mat_ideal_two_elt(nf,x));
1159 : }
1160 : else
1161 : {
1162 140786 : if (is_pm1(yZ)) return gcopy(x);
1163 35992 : z = idealHNF_mul_two(nf, x, mat_ideal_two_elt(nf,y));
1164 : }
1165 : }
1166 267825 : return z;
1167 : }
1168 :
1169 : /* operations on elements in factored form */
1170 :
1171 : GEN
1172 199533 : famat_mul_shallow(GEN f, GEN g)
1173 : {
1174 199533 : if (typ(f) != t_MAT) f = to_famat_shallow(f,gen_1);
1175 199533 : if (typ(g) != t_MAT) g = to_famat_shallow(g,gen_1);
1176 199533 : if (lgcols(f) == 1) return g;
1177 147732 : if (lgcols(g) == 1) return f;
1178 145836 : return mkmat2(shallowconcat(gel(f,1), gel(g,1)),
1179 145835 : shallowconcat(gel(f,2), gel(g,2)));
1180 : }
1181 : GEN
1182 88316 : famat_mulpow_shallow(GEN f, GEN g, GEN e)
1183 : {
1184 88316 : if (!signe(e)) return f;
1185 54610 : return famat_mul_shallow(f, famat_pow_shallow(g, e));
1186 : }
1187 :
1188 : GEN
1189 117984 : famat_mulpows_shallow(GEN f, GEN g, long e)
1190 : {
1191 117984 : if (e==0) return f;
1192 94450 : return famat_mul_shallow(f, famat_pows_shallow(g, e));
1193 : }
1194 :
1195 : GEN
1196 10304 : famat_div_shallow(GEN f, GEN g)
1197 10304 : { return famat_mul_shallow(f, famat_inv_shallow(g)); }
1198 :
1199 : GEN
1200 376251 : Z_to_famat(GEN x)
1201 : {
1202 : long k;
1203 376251 : if (equali1(x)) return trivial_fact();
1204 192262 : k = Z_isanypower(x, &x) ;
1205 192262 : return to_famat_shallow(x, k? utoi(k): gen_1);
1206 : }
1207 : GEN
1208 197053 : Q_to_famat(GEN x)
1209 : {
1210 197053 : if (typ(x) == t_INT) return Z_to_famat(x);
1211 179198 : return famat_div(Z_to_famat(gel(x,1)), Z_to_famat(gel(x,2)));
1212 : }
1213 : GEN
1214 0 : to_famat(GEN x, GEN y) { retmkmat2(mkcolcopy(x), mkcolcopy(y)); }
1215 : GEN
1216 2688368 : to_famat_shallow(GEN x, GEN y) { return mkmat2(mkcol(x), mkcol(y)); }
1217 :
1218 : /* concat the single elt x; not gconcat since x may be a t_COL */
1219 : static GEN
1220 151946 : append(GEN v, GEN x)
1221 : {
1222 151946 : long i, l = lg(v);
1223 151946 : GEN w = cgetg(l+1, typ(v));
1224 654040 : for (i=1; i<l; i++) gel(w,i) = gcopy(gel(v,i));
1225 151946 : gel(w,i) = gcopy(x); return w;
1226 : }
1227 : /* add x^1 to famat f */
1228 : static GEN
1229 158599 : famat_add(GEN f, GEN x)
1230 : {
1231 158599 : GEN h = cgetg(3,t_MAT);
1232 158599 : if (lgcols(f) == 1)
1233 : {
1234 13182 : gel(h,1) = mkcolcopy(x);
1235 13182 : gel(h,2) = mkcol(gen_1);
1236 : }
1237 : else
1238 : {
1239 145417 : gel(h,1) = append(gel(f,1), x);
1240 145417 : gel(h,2) = gconcat(gel(f,2), gen_1);
1241 : }
1242 158599 : return h;
1243 : }
1244 : /* add x^-1 to famat f */
1245 : static GEN
1246 20846 : famat_sub(GEN f, GEN x)
1247 : {
1248 20846 : GEN h = cgetg(3,t_MAT);
1249 20846 : if (lgcols(f) == 1)
1250 : {
1251 14317 : gel(h,1) = mkcolcopy(x);
1252 14317 : gel(h,2) = mkcol(gen_m1);
1253 : }
1254 : else
1255 : {
1256 6529 : gel(h,1) = append(gel(f,1), x);
1257 6529 : gel(h,2) = gconcat(gel(f,2), gen_m1);
1258 : }
1259 20846 : return h;
1260 : }
1261 :
1262 : GEN
1263 447353 : famat_mul(GEN f, GEN g)
1264 : {
1265 : GEN h;
1266 447353 : if (typ(g) != t_MAT) {
1267 30499 : if (typ(f) == t_MAT) return famat_add(f, g);
1268 0 : h = cgetg(3, t_MAT);
1269 0 : gel(h,1) = mkcol2(gcopy(f), gcopy(g));
1270 0 : gel(h,2) = mkcol2(gen_1, gen_1);
1271 : }
1272 416854 : if (typ(f) != t_MAT) return famat_add(g, f);
1273 288754 : if (lgcols(f) == 1) return gcopy(g);
1274 265433 : if (lgcols(g) == 1) return gcopy(f);
1275 259441 : h = cgetg(3,t_MAT);
1276 259441 : gel(h,1) = gconcat(gel(f,1), gel(g,1));
1277 259441 : gel(h,2) = gconcat(gel(f,2), gel(g,2));
1278 259441 : return h;
1279 : }
1280 :
1281 : GEN
1282 200051 : famat_div(GEN f, GEN g)
1283 : {
1284 : GEN h;
1285 200051 : if (typ(g) != t_MAT) {
1286 20804 : if (typ(f) == t_MAT) return famat_sub(f, g);
1287 0 : h = cgetg(3, t_MAT);
1288 0 : gel(h,1) = mkcol2(gcopy(f), gcopy(g));
1289 0 : gel(h,2) = mkcol2(gen_1, gen_m1);
1290 : }
1291 179247 : if (typ(f) != t_MAT) return famat_sub(g, f);
1292 179205 : if (lgcols(f) == 1) return famat_inv(g);
1293 260 : if (lgcols(g) == 1) return gcopy(f);
1294 260 : h = cgetg(3,t_MAT);
1295 260 : gel(h,1) = gconcat(gel(f,1), gel(g,1));
1296 260 : gel(h,2) = gconcat(gel(f,2), gneg(gel(g,2)));
1297 260 : return h;
1298 : }
1299 :
1300 : GEN
1301 22943 : famat_sqr(GEN f)
1302 : {
1303 : GEN h;
1304 22943 : if (typ(f) != t_MAT) return to_famat(f,gen_2);
1305 22943 : if (lgcols(f) == 1) return gcopy(f);
1306 13131 : h = cgetg(3,t_MAT);
1307 13131 : gel(h,1) = gcopy(gel(f,1));
1308 13131 : gel(h,2) = gmul2n(gel(f,2),1);
1309 13131 : return h;
1310 : }
1311 :
1312 : GEN
1313 26968 : famat_inv_shallow(GEN f)
1314 : {
1315 26968 : if (typ(f) != t_MAT) return to_famat_shallow(f,gen_m1);
1316 10444 : if (lgcols(f) == 1) return f;
1317 10444 : return mkmat2(gel(f,1), ZC_neg(gel(f,2)));
1318 : }
1319 : GEN
1320 199313 : famat_inv(GEN f)
1321 : {
1322 199313 : if (typ(f) != t_MAT) return to_famat(f,gen_m1);
1323 199313 : if (lgcols(f) == 1) return gcopy(f);
1324 180807 : retmkmat2(gcopy(gel(f,1)), ZC_neg(gel(f,2)));
1325 : }
1326 : GEN
1327 60642 : famat_pow(GEN f, GEN n)
1328 : {
1329 60642 : if (typ(f) != t_MAT) return to_famat(f,n);
1330 60642 : if (lgcols(f) == 1) return gcopy(f);
1331 60642 : retmkmat2(gcopy(gel(f,1)), ZC_Z_mul(gel(f,2),n));
1332 : }
1333 : GEN
1334 62037 : famat_pow_shallow(GEN f, GEN n)
1335 : {
1336 62037 : if (is_pm1(n)) return signe(n) > 0? f: famat_inv_shallow(f);
1337 35681 : if (typ(f) != t_MAT) return to_famat_shallow(f,n);
1338 7983 : if (lgcols(f) == 1) return f;
1339 6063 : return mkmat2(gel(f,1), ZC_Z_mul(gel(f,2),n));
1340 : }
1341 :
1342 : GEN
1343 122845 : famat_pows_shallow(GEN f, long n)
1344 : {
1345 122845 : if (n==1) return f;
1346 34706 : if (n==-1) return famat_inv_shallow(f);
1347 34699 : if (typ(f) != t_MAT) return to_famat_shallow(f, stoi(n));
1348 26474 : if (lgcols(f) == 1) return f;
1349 26474 : return mkmat2(gel(f,1), ZC_z_mul(gel(f,2),n));
1350 : }
1351 :
1352 : GEN
1353 0 : famat_Z_gcd(GEN M, GEN n)
1354 : {
1355 0 : pari_sp av=avma;
1356 0 : long i, j, l=lgcols(M);
1357 0 : GEN F=cgetg(3,t_MAT);
1358 0 : gel(F,1)=cgetg(l,t_COL);
1359 0 : gel(F,2)=cgetg(l,t_COL);
1360 0 : for (i=1, j=1; i<l; i++)
1361 : {
1362 0 : GEN p = gcoeff(M,i,1);
1363 0 : GEN e = gminsg(Z_pval(n,p),gcoeff(M,i,2));
1364 0 : if (signe(e))
1365 : {
1366 0 : gcoeff(F,j,1)=p;
1367 0 : gcoeff(F,j,2)=e;
1368 0 : j++;
1369 : }
1370 : }
1371 0 : setlg(gel(F,1),j); setlg(gel(F,2),j);
1372 0 : return gerepilecopy(av,F);
1373 : }
1374 :
1375 : /* x assumed to be a t_MATs (factorization matrix), or compatible with
1376 : * the element_* functions. */
1377 : static GEN
1378 33891 : ext_sqr(GEN nf, GEN x)
1379 33891 : { return (typ(x)==t_MAT)? famat_sqr(x): nfsqr(nf, x); }
1380 : static GEN
1381 60758 : ext_mul(GEN nf, GEN x, GEN y)
1382 60758 : { return (typ(x)==t_MAT)? famat_mul(x,y): nfmul(nf, x, y); }
1383 : static GEN
1384 20368 : ext_inv(GEN nf, GEN x)
1385 20368 : { return (typ(x)==t_MAT)? famat_inv(x): nfinv(nf, x); }
1386 : static GEN
1387 0 : ext_pow(GEN nf, GEN x, GEN n)
1388 0 : { return (typ(x)==t_MAT)? famat_pow(x,n): nfpow(nf, x, n); }
1389 :
1390 : GEN
1391 0 : famat_to_nf(GEN nf, GEN f)
1392 : {
1393 : GEN t, x, e;
1394 : long i;
1395 0 : if (lgcols(f) == 1) return gen_1;
1396 0 : x = gel(f,1);
1397 0 : e = gel(f,2);
1398 0 : t = nfpow(nf, gel(x,1), gel(e,1));
1399 0 : for (i=lg(x)-1; i>1; i--)
1400 0 : t = nfmul(nf, t, nfpow(nf, gel(x,i), gel(e,i)));
1401 0 : return t;
1402 : }
1403 :
1404 : GEN
1405 0 : famat_idealfactor(GEN nf, GEN x)
1406 : {
1407 : long i, l;
1408 0 : GEN g = gel(x,1), e = gel(x,2), h = cgetg_copy(g, &l);
1409 0 : for (i = 1; i < l; i++) gel(h,i) = idealfactor(nf, gel(g,i));
1410 0 : h = famat_reduce(famatV_factorback(h,e));
1411 0 : return sort_factor(h, (void*)&cmp_prime_ideal, &cmp_nodata);
1412 : }
1413 :
1414 : GEN
1415 295256 : famat_reduce(GEN fa)
1416 : {
1417 : GEN E, G, L, g, e;
1418 : long i, k, l;
1419 :
1420 295256 : if (typ(fa) != t_MAT || lgcols(fa) == 1) return fa;
1421 284537 : g = gel(fa,1); l = lg(g);
1422 284537 : e = gel(fa,2);
1423 284537 : L = gen_indexsort(g, (void*)&cmp_universal, &cmp_nodata);
1424 284537 : G = cgetg(l, t_COL);
1425 284537 : E = cgetg(l, t_COL);
1426 : /* merge */
1427 2293158 : for (k=i=1; i<l; i++,k++)
1428 : {
1429 2008620 : gel(G,k) = gel(g,L[i]);
1430 2008620 : gel(E,k) = gel(e,L[i]);
1431 2008620 : if (k > 1 && gidentical(gel(G,k), gel(G,k-1)))
1432 : {
1433 838424 : gel(E,k-1) = addii(gel(E,k), gel(E,k-1));
1434 838425 : k--;
1435 : }
1436 : }
1437 : /* kill 0 exponents */
1438 284538 : l = k;
1439 1454735 : for (k=i=1; i<l; i++)
1440 1170197 : if (!gequal0(gel(E,i)))
1441 : {
1442 1146655 : gel(G,k) = gel(G,i);
1443 1146655 : gel(E,k) = gel(E,i); k++;
1444 : }
1445 284538 : setlg(G, k);
1446 284538 : setlg(E, k); return mkmat2(G,E);
1447 : }
1448 : GEN
1449 63 : matreduce(GEN f)
1450 63 : { pari_sp av = avma;
1451 63 : switch(typ(f))
1452 : {
1453 21 : case t_VEC: case t_COL:
1454 : {
1455 21 : GEN e; f = vec_reduce(f, &e); settyp(f, t_COL);
1456 21 : return gerepilecopy(av, mkmat2(f, zc_to_ZC(e)));
1457 : }
1458 35 : case t_MAT:
1459 35 : if (lg(f) == 3) break;
1460 : default:
1461 14 : pari_err_TYPE("matreduce", f);
1462 : }
1463 28 : if (typ(gel(f,1)) == t_VECSMALL)
1464 0 : f = famatsmall_reduce(f);
1465 : else
1466 : {
1467 28 : if (!RgV_is_ZV(gel(f,2))) pari_err_TYPE("matreduce",f);
1468 21 : f = famat_reduce(f);
1469 : }
1470 21 : return gerepilecopy(av, f);
1471 : }
1472 :
1473 : GEN
1474 176591 : famatsmall_reduce(GEN fa)
1475 : {
1476 : GEN E, G, L, g, e;
1477 : long i, k, l;
1478 176591 : if (lgcols(fa) == 1) return fa;
1479 176591 : g = gel(fa,1); l = lg(g);
1480 176591 : e = gel(fa,2);
1481 176591 : L = vecsmall_indexsort(g);
1482 176592 : G = cgetg(l, t_VECSMALL);
1483 176592 : E = cgetg(l, t_VECSMALL);
1484 : /* merge */
1485 484518 : for (k=i=1; i<l; i++,k++)
1486 : {
1487 307926 : G[k] = g[L[i]];
1488 307926 : E[k] = e[L[i]];
1489 307926 : if (k > 1 && G[k] == G[k-1])
1490 : {
1491 7989 : E[k-1] += E[k];
1492 7989 : k--;
1493 : }
1494 : }
1495 : /* kill 0 exponents */
1496 176592 : l = k;
1497 476529 : for (k=i=1; i<l; i++)
1498 299937 : if (E[i])
1499 : {
1500 296268 : G[k] = G[i];
1501 296268 : E[k] = E[i]; k++;
1502 : }
1503 176592 : setlg(G, k);
1504 176592 : setlg(E, k); return mkmat2(G,E);
1505 : }
1506 :
1507 : GEN
1508 53614 : famat_remove_trivial(GEN fa)
1509 : {
1510 53614 : GEN P, E, p = gel(fa,1), e = gel(fa,2);
1511 53614 : long j, k, l = lg(p);
1512 53614 : P = cgetg(l, t_COL);
1513 53614 : E = cgetg(l, t_COL);
1514 1762623 : for (j = k = 1; j < l; j++)
1515 1709010 : if (signe(gel(e,j))) { gel(P,k) = gel(p,j); gel(E,k++) = gel(e,j); }
1516 53613 : setlg(P, k); setlg(E, k); return mkmat2(P,E);
1517 : }
1518 :
1519 : GEN
1520 13759 : famatV_factorback(GEN v, GEN e)
1521 : {
1522 13759 : long i, l = lg(e);
1523 : GEN V;
1524 13759 : if (l == 1) return trivial_fact();
1525 13374 : V = signe(gel(e,1))? famat_pow_shallow(gel(v,1), gel(e,1)): trivial_fact();
1526 56649 : for (i = 2; i < l; i++) V = famat_mulpow_shallow(V, gel(v,i), gel(e,i));
1527 13374 : return V;
1528 : }
1529 :
1530 : GEN
1531 46529 : famatV_zv_factorback(GEN v, GEN e)
1532 : {
1533 46529 : long i, l = lg(e);
1534 : GEN V;
1535 46529 : if (l == 1) return trivial_fact();
1536 44142 : V = uel(e,1)? famat_pows_shallow(gel(v,1), uel(e,1)): trivial_fact();
1537 138677 : for (i = 2; i < l; i++) V = famat_mulpows_shallow(V, gel(v,i), uel(e,i));
1538 44142 : return V;
1539 : }
1540 :
1541 : GEN
1542 568118 : ZM_famat_limit(GEN fa, GEN limit)
1543 : {
1544 : pari_sp av;
1545 : GEN E, G, g, e, r;
1546 : long i, k, l, n, lG;
1547 :
1548 568118 : if (lgcols(fa) == 1) return fa;
1549 568111 : g = gel(fa,1); l = lg(g);
1550 568111 : e = gel(fa,2);
1551 1137370 : for(n=0, i=1; i<l; i++)
1552 569259 : if (cmpii(gel(g,i),limit)<=0) n++;
1553 568111 : lG = n<l-1 ? n+2 : n+1;
1554 568111 : G = cgetg(lG, t_COL);
1555 568111 : E = cgetg(lG, t_COL);
1556 568111 : av = avma;
1557 1137370 : for (i=1, k=1, r = gen_1; i<l; i++)
1558 : {
1559 569259 : if (cmpii(gel(g,i),limit)<=0)
1560 : {
1561 569126 : gel(G,k) = gel(g,i);
1562 569126 : gel(E,k) = gel(e,i);
1563 569126 : k++;
1564 133 : } else r = mulii(r, powii(gel(g,i), gel(e,i)));
1565 : }
1566 568111 : if (k<i)
1567 : {
1568 133 : gel(G, k) = gerepileuptoint(av, r);
1569 133 : gel(E, k) = gen_1;
1570 : }
1571 568111 : return mkmat2(G,E);
1572 : }
1573 :
1574 : /* assume pr has degree 1 and coprime to Q_denom(x) */
1575 : static GEN
1576 122430 : to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1577 : {
1578 122430 : GEN d, r, p = modpr_get_p(modpr);
1579 122430 : x = nf_to_scalar_or_basis(nf,x);
1580 122430 : if (typ(x) != t_COL) return Rg_to_Fp(x,p);
1581 120792 : x = Q_remove_denom(x, &d);
1582 120792 : r = zk_to_Fq(x, modpr);
1583 120792 : if (d) r = Fp_div(r, d, p);
1584 120792 : return r;
1585 : }
1586 :
1587 : /* pr coprime to all denominators occurring in x */
1588 : static GEN
1589 693 : famat_to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1590 : {
1591 693 : GEN p = modpr_get_p(modpr);
1592 693 : GEN t = NULL, g = gel(x,1), e = gel(x,2), q = subiu(p,1);
1593 693 : long i, l = lg(g);
1594 4571 : for (i = 1; i < l; i++)
1595 : {
1596 3878 : GEN n = modii(gel(e,i), q);
1597 3878 : if (signe(n))
1598 : {
1599 3871 : GEN h = to_Fp_coprime(nf, gel(g,i), modpr);
1600 3871 : h = Fp_pow(h, n, p);
1601 3871 : t = t? Fp_mul(t, h, p): h;
1602 : }
1603 : }
1604 693 : return t? modii(t, p): gen_1;
1605 : }
1606 :
1607 : /* cf famat_to_nf_modideal_coprime, modpr attached to prime of degree 1 */
1608 : GEN
1609 119252 : nf_to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1610 : {
1611 693 : return typ(x)==t_MAT? famat_to_Fp_coprime(nf, x, modpr)
1612 119945 : : to_Fp_coprime(nf, x, modpr);
1613 : }
1614 :
1615 : static long
1616 4057092 : zk_pvalrem(GEN x, GEN p, GEN *py)
1617 4057092 : { return (typ(x) == t_INT)? Z_pvalrem(x, p, py): ZV_pvalrem(x, p, py); }
1618 : /* x a QC or Q. Return a ZC or Z, whose content is coprime to Z. Set v, dx
1619 : * such that x = p^v (newx / dx); dx = NULL if 1 */
1620 : static GEN
1621 4102919 : nf_remove_denom_p(GEN nf, GEN x, GEN p, GEN *pdx, long *pv)
1622 : {
1623 : long vcx;
1624 : GEN dx;
1625 4102919 : x = nf_to_scalar_or_basis(nf, x);
1626 4102919 : x = Q_remove_denom(x, &dx);
1627 4102920 : if (dx)
1628 : {
1629 61231 : vcx = - Z_pvalrem(dx, p, &dx);
1630 61231 : if (!vcx) vcx = zk_pvalrem(x, p, &x);
1631 61231 : if (isint1(dx)) dx = NULL;
1632 : }
1633 : else
1634 : {
1635 4041689 : vcx = zk_pvalrem(x, p, &x);
1636 4041693 : dx = NULL;
1637 : }
1638 4102924 : *pv = vcx;
1639 4102924 : *pdx = dx; return x;
1640 : }
1641 : /* x = b^e/p^(e-1) in Z_K; x = 0 mod p/pr^e, (x,pr) = 1. Return NULL
1642 : * if p inert (instead of 1) */
1643 : static GEN
1644 88856 : p_makecoprime(GEN pr)
1645 : {
1646 88856 : GEN B = pr_get_tau(pr), b;
1647 : long i, e;
1648 :
1649 88856 : if (typ(B) == t_INT) return NULL;
1650 66099 : b = gel(B,1); /* B = multiplication table by b */
1651 66099 : e = pr_get_e(pr);
1652 66099 : if (e == 1) return b;
1653 : /* one could also divide (exactly) by p in each iteration */
1654 41965 : for (i = 1; i < e; i++) b = ZM_ZC_mul(B, b);
1655 20469 : return ZC_Z_divexact(b, powiu(pr_get_p(pr), e-1));
1656 : }
1657 :
1658 : /* Compute A = prod g[i]^e[i] mod pr^k, assuming (A, pr) = 1.
1659 : * Method: modify each g[i] so that it becomes coprime to pr,
1660 : * g[i] *= (b/p)^v_pr(g[i]), where b/p = pr^(-1) times something integral
1661 : * and prime to p; globally, we multiply by (b/p)^v_pr(A) = 1.
1662 : * Optimizations:
1663 : * 1) remove all powers of p from contents, and consider extra generator p^vp;
1664 : * modified as p * (b/p)^e = b^e / p^(e-1)
1665 : * 2) remove denominators, coprime to p, by multiplying by inverse mod prk\cap Z
1666 : *
1667 : * EX = multiple of exponent of (O_K / pr^k)^* used to reduce the product in
1668 : * case the e[i] are large */
1669 : GEN
1670 2003515 : famat_makecoprime(GEN nf, GEN g, GEN e, GEN pr, GEN prk, GEN EX)
1671 : {
1672 2003515 : GEN G, E, t, vp = NULL, p = pr_get_p(pr), prkZ = gcoeff(prk, 1,1);
1673 2003514 : long i, l = lg(g);
1674 :
1675 2003514 : G = cgetg(l+1, t_VEC);
1676 2003530 : E = cgetg(l+1, t_VEC); /* l+1: room for "modified p" */
1677 6106424 : for (i=1; i < l; i++)
1678 : {
1679 : long vcx;
1680 4102919 : GEN dx, x = nf_remove_denom_p(nf, gel(g,i), p, &dx, &vcx);
1681 4102921 : if (vcx) /* = v_p(content(g[i])) */
1682 : {
1683 136044 : GEN a = mulsi(vcx, gel(e,i));
1684 136046 : vp = vp? addii(vp, a): a;
1685 : }
1686 : /* x integral, content coprime to p; dx coprime to p */
1687 4102922 : if (typ(x) == t_INT)
1688 : { /* x coprime to p, hence to pr */
1689 1109160 : x = modii(x, prkZ);
1690 1109159 : if (dx) x = Fp_div(x, dx, prkZ);
1691 : }
1692 : else
1693 : {
1694 2993762 : (void)ZC_nfvalrem(x, pr, &x); /* x *= (b/p)^v_pr(x) */
1695 2993732 : x = ZC_hnfrem(FpC_red(x,prkZ), prk);
1696 2993731 : if (dx) x = FpC_Fp_mul(x, Fp_inv(dx,prkZ), prkZ);
1697 : }
1698 4102873 : gel(G,i) = x;
1699 4102873 : gel(E,i) = gel(e,i);
1700 : }
1701 :
1702 2003505 : t = vp? p_makecoprime(pr): NULL;
1703 2003510 : if (!t)
1704 : { /* no need for extra generator */
1705 1937488 : setlg(G,l);
1706 1937490 : setlg(E,l);
1707 : }
1708 : else
1709 : {
1710 66022 : gel(G,i) = FpC_red(t, prkZ);
1711 66022 : gel(E,i) = vp;
1712 : }
1713 2003515 : return famat_to_nf_modideal_coprime(nf, G, E, prk, EX);
1714 : }
1715 :
1716 : /* simplified version of famat_makecoprime for X = SUnits[1] */
1717 : GEN
1718 98 : sunits_makecoprime(GEN X, GEN pr, GEN prk)
1719 : {
1720 98 : GEN G, p = pr_get_p(pr), prkZ = gcoeff(prk,1,1);
1721 98 : long i, l = lg(X);
1722 :
1723 98 : G = cgetg(l, t_VEC);
1724 9093 : for (i = 1; i < l; i++)
1725 : {
1726 8995 : GEN x = gel(X,i);
1727 8995 : if (typ(x) == t_INT) /* a prime */
1728 1421 : x = equalii(x,p)? p_makecoprime(pr): modii(x, prkZ);
1729 : else
1730 : {
1731 7574 : (void)ZC_nfvalrem(x, pr, &x); /* x *= (b/p)^v_pr(x) */
1732 7574 : x = ZC_hnfrem(FpC_red(x,prkZ), prk);
1733 : }
1734 8995 : gel(G,i) = x;
1735 : }
1736 98 : return G;
1737 : }
1738 :
1739 : /* prod g[i]^e[i] mod bid, assume (g[i], id) = 1 and 1 < lg(g) <= lg(e) */
1740 : GEN
1741 20118 : famat_to_nf_moddivisor(GEN nf, GEN g, GEN e, GEN bid)
1742 : {
1743 20118 : GEN t, cyc = bid_get_cyc(bid);
1744 20118 : if (lg(cyc) == 1)
1745 0 : t = gen_1;
1746 : else
1747 20118 : t = famat_to_nf_modideal_coprime(nf, g, e, bid_get_ideal(bid),
1748 : cyc_get_expo(cyc));
1749 20118 : return set_sign_mod_divisor(nf, mkmat2(g,e), t, bid_get_sarch(bid));
1750 : }
1751 :
1752 : GEN
1753 15989472 : vecmul(GEN x, GEN y)
1754 : {
1755 15989472 : if (!is_vec_t(typ(x))) return gmul(x,y);
1756 3521079 : pari_APPLY_same(vecmul(gel(x,i), gel(y,i)))
1757 : }
1758 :
1759 : GEN
1760 193473 : vecsqr(GEN x)
1761 : {
1762 193473 : if (!is_vec_t(typ(x))) return gsqr(x);
1763 54096 : pari_APPLY_same(vecsqr(gel(x,i)))
1764 : }
1765 :
1766 : GEN
1767 770 : vecinv(GEN x)
1768 : {
1769 770 : if (!is_vec_t(typ(x))) return ginv(x);
1770 0 : pari_APPLY_same(vecinv(gel(x,i)))
1771 : }
1772 :
1773 : GEN
1774 0 : vecpow(GEN x, GEN n)
1775 : {
1776 0 : if (!is_vec_t(typ(x))) return powgi(x,n);
1777 0 : pari_APPLY_same(vecpow(gel(x,i), n))
1778 : }
1779 :
1780 : GEN
1781 903 : vecdiv(GEN x, GEN y)
1782 : {
1783 903 : if (!is_vec_t(typ(x))) return gdiv(x,y);
1784 903 : pari_APPLY_same(vecdiv(gel(x,i), gel(y,i)))
1785 : }
1786 :
1787 : /* A ideal as a square t_MAT */
1788 : static GEN
1789 304682 : idealmulelt(GEN nf, GEN x, GEN A)
1790 : {
1791 : long i, lx;
1792 : GEN dx, dA, D;
1793 304682 : if (lg(A) == 1) return cgetg(1, t_MAT);
1794 304682 : x = nf_to_scalar_or_basis(nf,x);
1795 304682 : if (typ(x) != t_COL)
1796 99296 : return isintzero(x)? cgetg(1,t_MAT): RgM_Rg_mul(A, Q_abs_shallow(x));
1797 205386 : x = Q_remove_denom(x, &dx);
1798 205386 : A = Q_remove_denom(A, &dA);
1799 205386 : x = zk_multable(nf, x);
1800 205386 : D = mulii(zkmultable_capZ(x), gcoeff(A,1,1));
1801 205386 : x = zkC_multable_mul(A, x);
1802 205386 : settyp(x, t_MAT); lx = lg(x);
1803 : /* x may contain scalars (at most 1 since the ideal is nonzero)*/
1804 782611 : for (i=1; i<lx; i++)
1805 591946 : if (typ(gel(x,i)) == t_INT)
1806 : {
1807 14721 : if (i > 1) swap(gel(x,1), gel(x,i)); /* help HNF */
1808 14721 : gel(x,1) = scalarcol_shallow(gel(x,1), lx-1);
1809 14721 : break;
1810 : }
1811 205386 : x = ZM_hnfmodid(x, D);
1812 205386 : dx = mul_denom(dx,dA);
1813 205386 : return dx? gdiv(x,dx): x;
1814 : }
1815 :
1816 : /* nf a true nf, tx <= ty */
1817 : static GEN
1818 575321 : idealmul_aux(GEN nf, GEN x, GEN y, long tx, long ty)
1819 : {
1820 : GEN z, cx, cy;
1821 575321 : switch(tx)
1822 : {
1823 361671 : case id_PRINCIPAL:
1824 361671 : switch(ty)
1825 : {
1826 56597 : case id_PRINCIPAL:
1827 56597 : return idealhnf_principal(nf, nfmul(nf,x,y));
1828 392 : case id_PRIME:
1829 : {
1830 392 : GEN p = pr_get_p(y), pi = pr_get_gen(y), cx;
1831 392 : if (pr_is_inert(y)) return RgM_Rg_mul(idealhnf_principal(nf,x),p);
1832 :
1833 217 : x = nf_to_scalar_or_basis(nf, x);
1834 217 : switch(typ(x))
1835 : {
1836 203 : case t_INT:
1837 203 : if (!signe(x)) return cgetg(1,t_MAT);
1838 203 : return ZM_Z_mul(pr_hnf(nf,y), absi_shallow(x));
1839 7 : case t_FRAC:
1840 7 : return RgM_Rg_mul(pr_hnf(nf,y), Q_abs_shallow(x));
1841 : }
1842 : /* t_COL */
1843 7 : x = Q_primitive_part(x, &cx);
1844 7 : x = zk_multable(nf, x);
1845 7 : z = shallowconcat(ZM_Z_mul(x,p), ZM_ZC_mul(x,pi));
1846 7 : z = ZM_hnfmodid(z, mulii(p, zkmultable_capZ(x)));
1847 7 : return cx? ZM_Q_mul(z, cx): z;
1848 : }
1849 304682 : default: /* id_MAT */
1850 304682 : return idealmulelt(nf, x,y);
1851 : }
1852 42628 : case id_PRIME:
1853 42628 : if (ty==id_PRIME)
1854 4340 : { y = pr_hnf(nf,y); cy = NULL; }
1855 : else
1856 38288 : y = Q_primitive_part(y, &cy);
1857 42628 : y = idealHNF_mul_two(nf,y,x);
1858 42628 : return cy? ZM_Q_mul(y,cy): y;
1859 :
1860 171022 : default: /* id_MAT */
1861 : {
1862 171022 : long N = nf_get_degree(nf);
1863 171022 : if (lg(x)-1 != N || lg(y)-1 != N) pari_err_DIM("idealmul");
1864 171008 : x = Q_primitive_part(x, &cx);
1865 171008 : y = Q_primitive_part(y, &cy); cx = mul_content(cx,cy);
1866 171008 : y = idealHNF_mul(nf,x,y);
1867 171008 : return cx? ZM_Q_mul(y,cx): y;
1868 : }
1869 : }
1870 : }
1871 :
1872 : /* output the ideal product x.y */
1873 : GEN
1874 575321 : idealmul(GEN nf, GEN x, GEN y)
1875 : {
1876 : pari_sp av;
1877 : GEN res, ax, ay, z;
1878 575321 : long tx = idealtyp(&x,&ax);
1879 575321 : long ty = idealtyp(&y,&ay), f;
1880 575321 : if (tx>ty) { swap(ax,ay); swap(x,y); lswap(tx,ty); }
1881 575321 : f = (ax||ay); res = f? cgetg(3,t_VEC): NULL; /*product is an extended ideal*/
1882 575321 : av = avma;
1883 575321 : z = gerepileupto(av, idealmul_aux(checknf(nf), x,y, tx,ty));
1884 575307 : if (!f) return z;
1885 28015 : if (ax && ay)
1886 26531 : ax = ext_mul(nf, ax, ay);
1887 : else
1888 1484 : ax = gcopy(ax? ax: ay);
1889 28015 : gel(res,1) = z; gel(res,2) = ax; return res;
1890 : }
1891 :
1892 : /* Return x, integral in 2-elt form, such that pr^2 = c * x. cf idealpowprime
1893 : * nf = true nf */
1894 : static GEN
1895 281915 : idealsqrprime(GEN nf, GEN pr, GEN *pc)
1896 : {
1897 281915 : GEN p = pr_get_p(pr), q, gen;
1898 281914 : long e = pr_get_e(pr), f = pr_get_f(pr);
1899 :
1900 281915 : q = (e == 1)? sqri(p): p;
1901 281909 : if (e <= 2 && e * f == nf_get_degree(nf))
1902 : { /* pr^e = (p) */
1903 45269 : *pc = q;
1904 45269 : return mkvec2(gen_1,gen_0);
1905 : }
1906 236638 : gen = nfsqr(nf, pr_get_gen(pr));
1907 236643 : gen = FpC_red(gen, q);
1908 236641 : *pc = NULL;
1909 236641 : return mkvec2(q, gen);
1910 : }
1911 : /* cf idealpow_aux */
1912 : static GEN
1913 38560 : idealsqr_aux(GEN nf, GEN x, long tx)
1914 : {
1915 38560 : GEN T = nf_get_pol(nf), m, cx, a, alpha;
1916 38560 : long N = degpol(T);
1917 38560 : switch(tx)
1918 : {
1919 84 : case id_PRINCIPAL:
1920 84 : return idealhnf_principal(nf, nfsqr(nf,x));
1921 10773 : case id_PRIME:
1922 10773 : if (pr_is_inert(x)) return scalarmat(sqri(gel(x,1)), N);
1923 10605 : x = idealsqrprime(nf, x, &cx);
1924 10605 : x = idealhnf_two(nf,x);
1925 10605 : return cx? ZM_Z_mul(x, cx): x;
1926 27703 : default:
1927 27703 : x = Q_primitive_part(x, &cx);
1928 27703 : a = mat_ideal_two_elt(nf,x); alpha = gel(a,2); a = gel(a,1);
1929 27703 : alpha = nfsqr(nf,alpha);
1930 27703 : m = zk_scalar_or_multable(nf, alpha);
1931 27703 : if (typ(m) == t_INT) {
1932 1635 : x = gcdii(sqri(a), m);
1933 1635 : if (cx) x = gmul(x, gsqr(cx));
1934 1635 : x = scalarmat(x, N);
1935 : }
1936 : else
1937 : { /* could use gcdii(sqri(a), zkmultable_capZ(m)), but costly */
1938 26068 : x = ZM_hnfmodid(m, sqri(a));
1939 26068 : if (cx) cx = gsqr(cx);
1940 26068 : if (cx) x = ZM_Q_mul(x, cx);
1941 : }
1942 27703 : return x;
1943 : }
1944 : }
1945 : GEN
1946 38560 : idealsqr(GEN nf, GEN x)
1947 : {
1948 : pari_sp av;
1949 : GEN res, ax, z;
1950 38560 : long tx = idealtyp(&x,&ax);
1951 38560 : res = ax? cgetg(3,t_VEC): NULL; /*product is an extended ideal*/
1952 38560 : av = avma;
1953 38560 : z = gerepileupto(av, idealsqr_aux(checknf(nf), x, tx));
1954 38560 : if (!ax) return z;
1955 33891 : gel(res,1) = z;
1956 33891 : gel(res,2) = ext_sqr(nf, ax); return res;
1957 : }
1958 :
1959 : /* norm of an ideal */
1960 : GEN
1961 99885 : idealnorm(GEN nf, GEN x)
1962 : {
1963 : pari_sp av;
1964 : long tx;
1965 :
1966 99885 : switch(idealtyp(&x, NULL))
1967 : {
1968 952 : case id_PRIME: return pr_norm(x);
1969 9781 : case id_MAT: return RgM_det_triangular(x);
1970 : }
1971 : /* id_PRINCIPAL */
1972 89152 : nf = checknf(nf); av = avma;
1973 89152 : x = nfnorm(nf, x);
1974 89152 : tx = typ(x);
1975 89152 : if (tx == t_INT) return gerepileuptoint(av, absi(x));
1976 406 : if (tx != t_FRAC) pari_err_TYPE("idealnorm",x);
1977 406 : return gerepileupto(av, Q_abs(x));
1978 : }
1979 :
1980 : /* x \cap Z */
1981 : GEN
1982 2982 : idealdown(GEN nf, GEN x)
1983 : {
1984 2982 : pari_sp av = avma;
1985 : GEN y, c;
1986 2982 : switch(idealtyp(&x, NULL))
1987 : {
1988 7 : case id_PRIME: return icopy(pr_get_p(x));
1989 2107 : case id_MAT: return gcopy(gcoeff(x,1,1));
1990 : }
1991 : /* id_PRINCIPAL */
1992 868 : nf = checknf(nf); av = avma;
1993 868 : x = nf_to_scalar_or_basis(nf, x);
1994 868 : if (is_rational_t(typ(x))) return Q_abs(x);
1995 14 : x = Q_primitive_part(x, &c);
1996 14 : y = zkmultable_capZ(zk_multable(nf, x));
1997 14 : return gerepilecopy(av, mul_content(c, y));
1998 : }
1999 :
2000 : /* true nf */
2001 : static GEN
2002 35 : idealismaximal_int(GEN nf, GEN p)
2003 : {
2004 : GEN L;
2005 35 : if (!BPSW_psp(p)) return NULL;
2006 70 : if (!dvdii(nf_get_index(nf), p) &&
2007 49 : !FpX_is_irred(FpX_red(nf_get_pol(nf),p), p)) return NULL;
2008 21 : L = idealprimedec(nf, p);
2009 21 : return lg(L) == 2? gel(L,1): NULL;
2010 : }
2011 : /* true nf */
2012 : static GEN
2013 21 : idealismaximal_mat(GEN nf, GEN x)
2014 : {
2015 : GEN p, c, L;
2016 : long i, l, f;
2017 21 : x = Q_primitive_part(x, &c);
2018 21 : p = gcoeff(x,1,1);
2019 21 : if (c)
2020 : {
2021 7 : if (typ(c) == t_FRAC || !equali1(p)) return NULL;
2022 7 : return idealismaximal_int(nf, c);
2023 : }
2024 14 : if (!BPSW_psp(p)) return NULL;
2025 14 : l = lg(x); f = 1;
2026 35 : for (i = 2; i < l; i++)
2027 : {
2028 21 : c = gcoeff(x,i,i);
2029 21 : if (equalii(c, p)) f++; else if (!equali1(c)) return NULL;
2030 : }
2031 14 : L = idealprimedec_limit_f(nf, p, f);
2032 28 : for (i = lg(L)-1; i; i--)
2033 : {
2034 28 : GEN pr = gel(L,i);
2035 28 : if (pr_get_f(pr) != f) break;
2036 28 : if (idealval(nf, x, pr) == 1) return pr;
2037 : }
2038 0 : return NULL;
2039 : }
2040 : /* true nf */
2041 : static GEN
2042 70 : idealismaximal_i(GEN nf, GEN x)
2043 : {
2044 : GEN L, p, pr, c;
2045 : long i, l;
2046 70 : switch(idealtyp(&x, NULL))
2047 : {
2048 7 : case id_PRIME: return x;
2049 21 : case id_MAT: return idealismaximal_mat(nf, x);
2050 : }
2051 : /* id_PRINCIPAL */
2052 42 : x = nf_to_scalar_or_basis(nf, x);
2053 42 : switch(typ(x))
2054 : {
2055 28 : case t_INT: return idealismaximal_int(nf, absi_shallow(x));
2056 0 : case t_FRAC: return NULL;
2057 : }
2058 14 : x = Q_primitive_part(x, &c);
2059 14 : if (c) return NULL;
2060 14 : p = zkmultable_capZ(zk_multable(nf, x));
2061 14 : if (!BPSW_psp(p)) return NULL;
2062 7 : L = idealprimedec(nf, p); l = lg(L); pr = NULL;
2063 21 : for (i = 1; i < l; i++)
2064 : {
2065 14 : long v = ZC_nfval(x, gel(L,i));
2066 14 : if (v > 1 || (v && pr)) return NULL;
2067 14 : pr = gel(L,i);
2068 : }
2069 7 : return pr;
2070 : }
2071 : GEN
2072 70 : idealismaximal(GEN nf, GEN x)
2073 : {
2074 70 : pari_sp av = avma;
2075 70 : x = idealismaximal_i(checknf(nf), x);
2076 70 : if (!x) { set_avma(av); return gen_0; }
2077 49 : return gerepilecopy(av, x);
2078 : }
2079 :
2080 : /* I^(-1) = { x \in K, Tr(x D^(-1) I) \in Z }, D different of K/Q
2081 : *
2082 : * nf[5][6] = pp( D^(-1) ) = pp( HNF( T^(-1) ) ), T = (Tr(wi wj))
2083 : * nf[5][7] = same in 2-elt form.
2084 : * Assume I integral. Return the integral ideal (I\cap Z) I^(-1) */
2085 : GEN
2086 217561 : idealHNF_inv_Z(GEN nf, GEN I)
2087 : {
2088 217561 : GEN J, dual, IZ = gcoeff(I,1,1); /* I \cap Z */
2089 217561 : if (isint1(IZ)) return matid(lg(I)-1);
2090 197702 : J = idealHNF_mul(nf,I, gmael(nf,5,7));
2091 : /* I in HNF, hence easily inverted; multiply by IZ to get integer coeffs
2092 : * missing content cancels while solving the linear equation */
2093 197704 : dual = shallowtrans( hnf_divscale(J, gmael(nf,5,6), IZ) );
2094 197704 : return ZM_hnfmodid(dual, IZ);
2095 : }
2096 : /* I HNF with rational coefficients (denominator d). */
2097 : GEN
2098 78908 : idealHNF_inv(GEN nf, GEN I)
2099 : {
2100 78908 : GEN J, IQ = gcoeff(I,1,1); /* I \cap Q; d IQ = dI \cap Z */
2101 78908 : J = idealHNF_inv_Z(nf, Q_remove_denom(I, NULL)); /* = (dI)^(-1) * (d IQ) */
2102 78908 : return equali1(IQ)? J: RgM_Rg_div(J, IQ);
2103 : }
2104 :
2105 : /* return p * P^(-1) [integral] */
2106 : GEN
2107 39362 : pr_inv_p(GEN pr)
2108 : {
2109 39362 : if (pr_is_inert(pr)) return matid(pr_get_f(pr));
2110 38578 : return ZM_hnfmodid(pr_get_tau(pr), pr_get_p(pr));
2111 : }
2112 : GEN
2113 18257 : pr_inv(GEN pr)
2114 : {
2115 18257 : GEN p = pr_get_p(pr);
2116 18257 : if (pr_is_inert(pr)) return scalarmat(ginv(p), pr_get_f(pr));
2117 17865 : return RgM_Rg_div(ZM_hnfmodid(pr_get_tau(pr),p), p);
2118 : }
2119 :
2120 : GEN
2121 149137 : idealinv(GEN nf, GEN x)
2122 : {
2123 : GEN res, ax;
2124 : pari_sp av;
2125 149137 : long tx = idealtyp(&x,&ax), N;
2126 :
2127 149137 : res = ax? cgetg(3,t_VEC): NULL;
2128 149137 : nf = checknf(nf); av = avma;
2129 149137 : N = nf_get_degree(nf);
2130 149137 : switch (tx)
2131 : {
2132 71784 : case id_MAT:
2133 71784 : if (lg(x)-1 != N) pari_err_DIM("idealinv");
2134 71784 : x = idealHNF_inv(nf,x); break;
2135 60376 : case id_PRINCIPAL:
2136 60376 : x = nf_to_scalar_or_basis(nf, x);
2137 60376 : if (typ(x) != t_COL)
2138 60327 : x = idealhnf_principal(nf,ginv(x));
2139 : else
2140 : { /* nfinv + idealhnf where we already know (x) \cap Z */
2141 : GEN c, d;
2142 49 : x = Q_remove_denom(x, &c);
2143 49 : x = zk_inv(nf, x);
2144 49 : x = Q_remove_denom(x, &d); /* true inverse is c/d * x */
2145 49 : if (!d) /* x and x^(-1) integral => x a unit */
2146 14 : x = c? scalarmat(c, N): matid(N);
2147 : else
2148 : {
2149 35 : c = c? gdiv(c,d): ginv(d);
2150 35 : x = zk_multable(nf, x);
2151 35 : x = ZM_Q_mul(ZM_hnfmodid(x,d), c);
2152 : }
2153 : }
2154 60376 : break;
2155 16977 : case id_PRIME:
2156 16977 : x = pr_inv(x); break;
2157 : }
2158 149137 : x = gerepileupto(av,x); if (!ax) return x;
2159 20368 : gel(res,1) = x;
2160 20368 : gel(res,2) = ext_inv(nf, ax); return res;
2161 : }
2162 :
2163 : /* write x = A/B, A,B coprime integral ideals */
2164 : GEN
2165 377910 : idealnumden(GEN nf, GEN x)
2166 : {
2167 377910 : pari_sp av = avma;
2168 : GEN x0, c, d, A, B, J;
2169 377910 : long tx = idealtyp(&x, NULL);
2170 377907 : nf = checknf(nf);
2171 377910 : switch (tx)
2172 : {
2173 7 : case id_PRIME:
2174 7 : retmkvec2(idealhnf(nf, x), gen_1);
2175 137521 : case id_PRINCIPAL:
2176 : {
2177 : GEN xZ, mx;
2178 137521 : x = nf_to_scalar_or_basis(nf, x);
2179 137521 : switch(typ(x))
2180 : {
2181 86198 : case t_INT: return gerepilecopy(av, mkvec2(absi_shallow(x),gen_1));
2182 2639 : case t_FRAC:return gerepilecopy(av, mkvec2(absi_shallow(gel(x,1)), gel(x,2)));
2183 : }
2184 : /* t_COL */
2185 48684 : x = Q_remove_denom(x, &d);
2186 48685 : if (!d) return gerepilecopy(av, mkvec2(idealhnf_shallow(nf, x), gen_1));
2187 105 : mx = zk_multable(nf, x);
2188 105 : xZ = zkmultable_capZ(mx);
2189 105 : x = ZM_hnfmodid(mx, xZ); /* principal ideal (x) */
2190 105 : x0 = mkvec2(xZ, mx); /* same, for fast multiplication */
2191 105 : break;
2192 : }
2193 240382 : default: /* id_MAT */
2194 : {
2195 240382 : long n = lg(x)-1;
2196 240382 : if (n == 0) return mkvec2(gen_0, gen_1);
2197 240382 : if (n != nf_get_degree(nf)) pari_err_DIM("idealnumden");
2198 240382 : x0 = x = Q_remove_denom(x, &d);
2199 240373 : if (!d) return gerepilecopy(av, mkvec2(x, gen_1));
2200 21 : break;
2201 : }
2202 : }
2203 126 : J = hnfmodid(x, d); /* = d/B */
2204 126 : c = gcoeff(J,1,1); /* (d/B) \cap Z, divides d */
2205 126 : B = idealHNF_inv_Z(nf, J); /* (d/B \cap Z) B/d */
2206 126 : if (!equalii(c,d)) B = ZM_Z_mul(B, diviiexact(d,c)); /* = B ! */
2207 126 : A = idealHNF_mul(nf, B, x0); /* d * (original x) * B = d A */
2208 126 : A = ZM_Z_divexact(A, d); /* = A ! */
2209 126 : return gerepilecopy(av, mkvec2(A, B));
2210 : }
2211 :
2212 : /* Return x, integral in 2-elt form, such that pr^n = c * x. Assume n != 0.
2213 : * nf = true nf */
2214 : static GEN
2215 1104810 : idealpowprime(GEN nf, GEN pr, GEN n, GEN *pc)
2216 : {
2217 1104810 : GEN p = pr_get_p(pr), q, gen;
2218 :
2219 1104802 : *pc = NULL;
2220 1104802 : if (is_pm1(n)) /* n = 1 special cased for efficiency */
2221 : {
2222 596734 : q = p;
2223 596734 : if (typ(pr_get_tau(pr)) == t_INT) /* inert */
2224 : {
2225 0 : *pc = (signe(n) >= 0)? p: ginv(p);
2226 0 : return mkvec2(gen_1,gen_0);
2227 : }
2228 596714 : if (signe(n) >= 0) gen = pr_get_gen(pr);
2229 : else
2230 : {
2231 155556 : gen = pr_get_tau(pr); /* possibly t_MAT */
2232 155577 : *pc = ginv(p);
2233 : }
2234 : }
2235 508146 : else if (equalis(n,2)) return idealsqrprime(nf, pr, pc);
2236 : else
2237 : {
2238 236836 : long e = pr_get_e(pr), f = pr_get_f(pr);
2239 236843 : GEN r, m = truedvmdis(n, e, &r);
2240 236838 : if (e * f == nf_get_degree(nf))
2241 : { /* pr^e = (p) */
2242 75661 : if (signe(m)) *pc = powii(p,m);
2243 75663 : if (!signe(r)) return mkvec2(gen_1,gen_0);
2244 36018 : q = p;
2245 36018 : gen = nfpow(nf, pr_get_gen(pr), r);
2246 : }
2247 : else
2248 : {
2249 161179 : m = absi_shallow(m);
2250 161180 : if (signe(r)) m = addiu(m,1);
2251 161180 : q = powii(p,m); /* m = ceil(|n|/e) */
2252 161181 : if (signe(n) >= 0) gen = nfpow(nf, pr_get_gen(pr), n);
2253 : else
2254 : {
2255 24272 : gen = pr_get_tau(pr);
2256 24272 : if (typ(gen) == t_MAT) gen = gel(gen,1);
2257 24272 : n = negi(n);
2258 24272 : gen = ZC_Z_divexact(nfpow(nf, gen, n), powii(p, subii(n,m)));
2259 24272 : *pc = ginv(q);
2260 : }
2261 : }
2262 197200 : gen = FpC_red(gen, q);
2263 : }
2264 793929 : return mkvec2(q, gen);
2265 : }
2266 :
2267 : /* True nf. x * pr^n. Assume x in HNF or scalar (possibly nonintegral) */
2268 : GEN
2269 729946 : idealmulpowprime(GEN nf, GEN x, GEN pr, GEN n)
2270 : {
2271 : GEN c, cx, y;
2272 729946 : long N = nf_get_degree(nf);
2273 :
2274 729922 : if (!signe(n)) return typ(x) == t_MAT? x: scalarmat_shallow(x, N);
2275 :
2276 : /* inert, special cased for efficiency */
2277 729915 : if (pr_is_inert(pr))
2278 : {
2279 75017 : GEN q = powii(pr_get_p(pr), n);
2280 72956 : return typ(x) == t_MAT? RgM_Rg_mul(x,q)
2281 147982 : : scalarmat_shallow(gmul(Q_abs(x),q), N);
2282 : }
2283 :
2284 654907 : y = idealpowprime(nf, pr, n, &c);
2285 654911 : if (typ(x) == t_MAT)
2286 651784 : { x = Q_primitive_part(x, &cx); if (is_pm1(gcoeff(x,1,1))) x = NULL; }
2287 : else
2288 3127 : { cx = x; x = NULL; }
2289 654785 : cx = mul_content(c,cx);
2290 654782 : if (x)
2291 489331 : x = idealHNF_mul_two(nf,x,y);
2292 : else
2293 165451 : x = idealhnf_two(nf,y);
2294 655098 : if (cx) x = ZM_Q_mul(x,cx);
2295 654894 : return x;
2296 : }
2297 : GEN
2298 9997 : idealdivpowprime(GEN nf, GEN x, GEN pr, GEN n)
2299 : {
2300 9997 : return idealmulpowprime(nf,x,pr, negi(n));
2301 : }
2302 :
2303 : /* nf = true nf */
2304 : static GEN
2305 825408 : idealpow_aux(GEN nf, GEN x, long tx, GEN n)
2306 : {
2307 825408 : GEN T = nf_get_pol(nf), m, cx, n1, a, alpha;
2308 825407 : long N = degpol(T), s = signe(n);
2309 825414 : if (!s) return matid(N);
2310 810130 : switch(tx)
2311 : {
2312 75171 : case id_PRINCIPAL:
2313 75171 : return idealhnf_principal(nf, nfpow(nf,x,n));
2314 541478 : case id_PRIME:
2315 541478 : if (pr_is_inert(x)) return scalarmat(powii(gel(x,1), n), N);
2316 449867 : x = idealpowprime(nf, x, n, &cx);
2317 449868 : x = idealhnf_two(nf,x);
2318 449875 : return cx? ZM_Q_mul(x, cx): x;
2319 193481 : default:
2320 193481 : if (is_pm1(n)) return (s < 0)? idealinv(nf, x): gcopy(x);
2321 69145 : n1 = (s < 0)? negi(n): n;
2322 :
2323 69145 : x = Q_primitive_part(x, &cx);
2324 69145 : a = mat_ideal_two_elt(nf,x); alpha = gel(a,2); a = gel(a,1);
2325 69145 : alpha = nfpow(nf,alpha,n1);
2326 69145 : m = zk_scalar_or_multable(nf, alpha);
2327 69145 : if (typ(m) == t_INT) {
2328 553 : x = gcdii(powii(a,n1), m);
2329 553 : if (s<0) x = ginv(x);
2330 553 : if (cx) x = gmul(x, powgi(cx,n));
2331 553 : x = scalarmat(x, N);
2332 : }
2333 : else
2334 : { /* could use gcdii(powii(a,n1), zkmultable_capZ(m)), but costly */
2335 68592 : x = ZM_hnfmodid(m, powii(a,n1));
2336 68592 : if (cx) cx = powgi(cx,n);
2337 68592 : if (s<0) {
2338 7 : GEN xZ = gcoeff(x,1,1);
2339 7 : cx = cx ? gdiv(cx, xZ): ginv(xZ);
2340 7 : x = idealHNF_inv_Z(nf,x);
2341 : }
2342 68592 : if (cx) x = ZM_Q_mul(x, cx);
2343 : }
2344 69145 : return x;
2345 : }
2346 : }
2347 :
2348 : /* raise the ideal x to the power n (in Z) */
2349 : GEN
2350 825410 : idealpow(GEN nf, GEN x, GEN n)
2351 : {
2352 : pari_sp av;
2353 : long tx;
2354 : GEN res, ax;
2355 :
2356 825410 : if (typ(n) != t_INT) pari_err_TYPE("idealpow",n);
2357 825410 : tx = idealtyp(&x,&ax);
2358 825412 : res = ax? cgetg(3,t_VEC): NULL;
2359 825412 : av = avma;
2360 825412 : x = gerepileupto(av, idealpow_aux(checknf(nf), x, tx, n));
2361 825415 : if (!ax) return x;
2362 0 : gel(res,1) = x;
2363 0 : gel(res,2) = ext_pow(nf, ax, n);
2364 0 : return res;
2365 : }
2366 :
2367 : /* Return ideal^e in number field nf. e is a C integer. */
2368 : GEN
2369 274577 : idealpows(GEN nf, GEN ideal, long e)
2370 : {
2371 274577 : long court[] = {evaltyp(t_INT) | _evallg(3),0,0};
2372 274577 : affsi(e,court); return idealpow(nf,ideal,court);
2373 : }
2374 :
2375 : static GEN
2376 28589 : _idealmulred(GEN nf, GEN x, GEN y)
2377 28589 : { return idealred(nf,idealmul(nf,x,y)); }
2378 : static GEN
2379 35676 : _idealsqrred(GEN nf, GEN x)
2380 35676 : { return idealred(nf,idealsqr(nf,x)); }
2381 : static GEN
2382 11382 : _mul(void *data, GEN x, GEN y) { return _idealmulred((GEN)data,x,y); }
2383 : static GEN
2384 35676 : _sqr(void *data, GEN x) { return _idealsqrred((GEN)data, x); }
2385 :
2386 : /* compute x^n (x ideal, n integer), reducing along the way */
2387 : GEN
2388 80202 : idealpowred(GEN nf, GEN x, GEN n)
2389 : {
2390 80202 : pari_sp av = avma, av2;
2391 : long s;
2392 : GEN y;
2393 :
2394 80202 : if (typ(n) != t_INT) pari_err_TYPE("idealpowred",n);
2395 80202 : s = signe(n); if (s == 0) return idealpow(nf,x,n);
2396 80202 : y = gen_pow_i(x, n, (void*)nf, &_sqr, &_mul);
2397 80202 : av2 = avma;
2398 80202 : if (s < 0) y = idealinv(nf,y);
2399 80201 : if (s < 0 || is_pm1(n)) y = idealred(nf,y);
2400 80202 : return avma == av2? gerepilecopy(av,y): gerepileupto(av,y);
2401 : }
2402 :
2403 : GEN
2404 17207 : idealmulred(GEN nf, GEN x, GEN y)
2405 : {
2406 17207 : pari_sp av = avma;
2407 17207 : return gerepileupto(av, _idealmulred(nf,x,y));
2408 : }
2409 :
2410 : long
2411 91 : isideal(GEN nf,GEN x)
2412 : {
2413 91 : long N, i, j, lx, tx = typ(x);
2414 : pari_sp av;
2415 : GEN T, xZ;
2416 :
2417 91 : nf = checknf(nf); T = nf_get_pol(nf); lx = lg(x);
2418 91 : if (tx==t_VEC && lx==3) { x = gel(x,1); tx = typ(x); lx = lg(x); }
2419 91 : switch(tx)
2420 : {
2421 14 : case t_INT: case t_FRAC: return 1;
2422 7 : case t_POL: return varn(x) == varn(T);
2423 7 : case t_POLMOD: return RgX_equal_var(T, gel(x,1));
2424 14 : case t_VEC: return get_prid(x)? 1 : 0;
2425 42 : case t_MAT: break;
2426 7 : default: return 0;
2427 : }
2428 42 : N = degpol(T);
2429 42 : if (lx-1 != N) return (lx == 1);
2430 28 : if (nbrows(x) != N) return 0;
2431 :
2432 28 : av = avma; x = Q_primpart(x);
2433 28 : if (!ZM_ishnf(x)) return 0;
2434 14 : xZ = gcoeff(x,1,1);
2435 21 : for (j=2; j<=N; j++)
2436 14 : if (!dvdii(xZ, gcoeff(x,j,j))) return gc_long(av,0);
2437 14 : for (i=2; i<=N; i++)
2438 14 : for (j=2; j<=N; j++)
2439 7 : if (! hnf_invimage(x, zk_ei_mul(nf,gel(x,i),j))) return gc_long(av,0);
2440 7 : return gc_long(av,1);
2441 : }
2442 :
2443 : GEN
2444 39792 : idealdiv(GEN nf, GEN x, GEN y)
2445 : {
2446 39792 : pari_sp av = avma, tetpil;
2447 39792 : GEN z = idealinv(nf,y);
2448 39792 : tetpil = avma; return gerepile(av,tetpil, idealmul(nf,x,z));
2449 : }
2450 :
2451 : /* This routine computes the quotient x/y of two ideals in the number field nf.
2452 : * It assumes that the quotient is an integral ideal. The idea is to find an
2453 : * ideal z dividing y such that gcd(Nx/Nz, Nz) = 1. Then
2454 : *
2455 : * x + (Nx/Nz) x
2456 : * ----------- = ---
2457 : * y + (Ny/Nz) y
2458 : *
2459 : * Proof: we can assume x and y are integral. Let p be any prime ideal
2460 : *
2461 : * If p | Nz, then it divides neither Nx/Nz nor Ny/Nz (since Nx/Nz is the
2462 : * product of the integers N(x/y) and N(y/z)). Both the numerator and the
2463 : * denominator on the left will be coprime to p. So will x/y, since x/y is
2464 : * assumed integral and its norm N(x/y) is coprime to p.
2465 : *
2466 : * If instead p does not divide Nz, then v_p (Nx/Nz) = v_p (Nx) >= v_p(x).
2467 : * Hence v_p (x + Nx/Nz) = v_p(x). Likewise for the denominators. QED.
2468 : *
2469 : * Peter Montgomery. July, 1994. */
2470 : static void
2471 7 : err_divexact(GEN x, GEN y)
2472 7 : { pari_err_DOMAIN("idealdivexact","denominator(x/y)", "!=",
2473 0 : gen_1,mkvec2(x,y)); }
2474 : GEN
2475 4404 : idealdivexact(GEN nf, GEN x0, GEN y0)
2476 : {
2477 4404 : pari_sp av = avma;
2478 : GEN x, y, xZ, yZ, Nx, Ny, Nz, cy, q, r;
2479 :
2480 4404 : nf = checknf(nf);
2481 4404 : x = idealhnf_shallow(nf, x0);
2482 4404 : y = idealhnf_shallow(nf, y0);
2483 4404 : if (lg(y) == 1) pari_err_INV("idealdivexact", y0);
2484 4397 : if (lg(x) == 1) { set_avma(av); return cgetg(1, t_MAT); } /* numerator is zero */
2485 4397 : y = Q_primitive_part(y, &cy);
2486 4397 : if (cy) x = RgM_Rg_div(x,cy);
2487 4397 : xZ = gcoeff(x,1,1); if (typ(xZ) != t_INT) err_divexact(x,y);
2488 4390 : yZ = gcoeff(y,1,1); if (isint1(yZ)) return gerepilecopy(av, x);
2489 2661 : Nx = idealnorm(nf,x);
2490 2661 : Ny = idealnorm(nf,y);
2491 2661 : if (typ(Nx) != t_INT) err_divexact(x,y);
2492 2661 : q = dvmdii(Nx,Ny, &r);
2493 2661 : if (signe(r)) err_divexact(x,y);
2494 2661 : if (is_pm1(q)) { set_avma(av); return matid(nf_get_degree(nf)); }
2495 : /* Find a norm Nz | Ny such that gcd(Nx/Nz, Nz) = 1 */
2496 485 : for (Nz = Ny;;) /* q = Nx/Nz */
2497 465 : {
2498 950 : GEN p1 = gcdii(Nz, q);
2499 950 : if (is_pm1(p1)) break;
2500 465 : Nz = diviiexact(Nz,p1);
2501 465 : q = mulii(q,p1);
2502 : }
2503 485 : xZ = gcoeff(x,1,1); q = gcdii(q, xZ);
2504 485 : if (!equalii(xZ,q))
2505 : { /* Replace x/y by x+(Nx/Nz) / y+(Ny/Nz) */
2506 335 : x = ZM_hnfmodid(x, q);
2507 : /* y reduced to unit ideal ? */
2508 335 : if (Nz == Ny) return gerepileupto(av, x);
2509 :
2510 118 : yZ = gcoeff(y,1,1); q = gcdii(diviiexact(Ny,Nz), yZ);
2511 118 : y = ZM_hnfmodid(y, q);
2512 : }
2513 268 : yZ = gcoeff(y,1,1);
2514 268 : y = idealHNF_mul(nf,x, idealHNF_inv_Z(nf,y));
2515 268 : return gerepileupto(av, ZM_Z_divexact(y, yZ));
2516 : }
2517 :
2518 : GEN
2519 21 : idealintersect(GEN nf, GEN x, GEN y)
2520 : {
2521 21 : pari_sp av = avma;
2522 : long lz, lx, i;
2523 : GEN z, dx, dy, xZ, yZ;;
2524 :
2525 21 : nf = checknf(nf);
2526 21 : x = idealhnf_shallow(nf,x);
2527 21 : y = idealhnf_shallow(nf,y);
2528 21 : if (lg(x) == 1 || lg(y) == 1) { set_avma(av); return cgetg(1,t_MAT); }
2529 14 : x = Q_remove_denom(x, &dx);
2530 14 : y = Q_remove_denom(y, &dy);
2531 14 : if (dx) y = ZM_Z_mul(y, dx);
2532 14 : if (dy) x = ZM_Z_mul(x, dy);
2533 14 : xZ = gcoeff(x,1,1);
2534 14 : yZ = gcoeff(y,1,1);
2535 14 : dx = mul_denom(dx,dy);
2536 14 : z = ZM_lll(shallowconcat(x,y), 0.99, LLL_KER); lz = lg(z);
2537 14 : lx = lg(x);
2538 63 : for (i=1; i<lz; i++) setlg(z[i], lx);
2539 14 : z = ZM_hnfmodid(ZM_mul(x,z), lcmii(xZ, yZ));
2540 14 : if (dx) z = RgM_Rg_div(z,dx);
2541 14 : return gerepileupto(av,z);
2542 : }
2543 :
2544 : /*******************************************************************/
2545 : /* */
2546 : /* T2-IDEAL REDUCTION */
2547 : /* */
2548 : /*******************************************************************/
2549 :
2550 : static GEN
2551 21 : chk_vdir(GEN nf, GEN vdir)
2552 : {
2553 21 : long i, l = lg(vdir);
2554 : GEN v;
2555 21 : if (l != lg(nf_get_roots(nf))) pari_err_DIM("idealred");
2556 14 : switch(typ(vdir))
2557 : {
2558 0 : case t_VECSMALL: return vdir;
2559 14 : case t_VEC: break;
2560 0 : default: pari_err_TYPE("idealred",vdir);
2561 : }
2562 14 : v = cgetg(l, t_VECSMALL);
2563 56 : for (i = 1; i < l; i++) v[i] = itos(gceil(gel(vdir,i)));
2564 14 : return v;
2565 : }
2566 :
2567 : static void
2568 12625 : twistG(GEN G, long r1, long i, long v)
2569 : {
2570 12625 : long j, lG = lg(G);
2571 12625 : if (i <= r1) {
2572 37233 : for (j=1; j<lG; j++) gcoeff(G,i,j) = gmul2n(gcoeff(G,i,j), v);
2573 : } else {
2574 578 : long k = (i<<1) - r1;
2575 4402 : for (j=1; j<lG; j++)
2576 : {
2577 3824 : gcoeff(G,k-1,j) = gmul2n(gcoeff(G,k-1,j), v);
2578 3824 : gcoeff(G,k ,j) = gmul2n(gcoeff(G,k ,j), v);
2579 : }
2580 : }
2581 12625 : }
2582 :
2583 : GEN
2584 138605 : nf_get_Gtwist(GEN nf, GEN vdir)
2585 : {
2586 : long i, l, v, r1;
2587 : GEN G;
2588 :
2589 138605 : if (!vdir) return nf_get_roundG(nf);
2590 21 : if (typ(vdir) == t_MAT)
2591 : {
2592 0 : long N = nf_get_degree(nf);
2593 0 : if (lg(vdir) != N+1 || lgcols(vdir) != N+1) pari_err_DIM("idealred");
2594 0 : return vdir;
2595 : }
2596 21 : vdir = chk_vdir(nf, vdir);
2597 14 : G = RgM_shallowcopy(nf_get_G(nf));
2598 14 : r1 = nf_get_r1(nf);
2599 14 : l = lg(vdir);
2600 56 : for (i=1; i<l; i++)
2601 : {
2602 42 : v = vdir[i]; if (!v) continue;
2603 42 : twistG(G, r1, i, v);
2604 : }
2605 14 : return RM_round_maxrank(G);
2606 : }
2607 : GEN
2608 12583 : nf_get_Gtwist1(GEN nf, long i)
2609 : {
2610 12583 : GEN G = RgM_shallowcopy( nf_get_G(nf) );
2611 12583 : long r1 = nf_get_r1(nf);
2612 12583 : twistG(G, r1, i, 10);
2613 12583 : return RM_round_maxrank(G);
2614 : }
2615 :
2616 : GEN
2617 96369 : RM_round_maxrank(GEN G0)
2618 : {
2619 96369 : long e, r = lg(G0)-1;
2620 96369 : pari_sp av = avma;
2621 96369 : for (e = 4; ; e <<= 1, set_avma(av))
2622 0 : {
2623 96369 : GEN G = gmul2n(G0, e), H = ground(G);
2624 96369 : if (ZM_rank(H) == r) return H; /* maximal rank ? */
2625 : }
2626 : }
2627 :
2628 : GEN
2629 138598 : idealred0(GEN nf, GEN I, GEN vdir)
2630 : {
2631 138598 : pari_sp av = avma;
2632 138598 : GEN G, aI, IZ, J, y, my, dyi, yi, c1 = NULL;
2633 : long N;
2634 :
2635 138598 : nf = checknf(nf);
2636 138598 : N = nf_get_degree(nf);
2637 : /* put first for sanity checks, unused when I obviously principal */
2638 138598 : G = nf_get_Gtwist(nf, vdir);
2639 138591 : switch (idealtyp(&I,&aI))
2640 : {
2641 37197 : case id_PRIME:
2642 37197 : if (pr_is_inert(I)) {
2643 655 : if (!aI) { set_avma(av); return matid(N); }
2644 655 : c1 = gel(I,1); I = matid(N);
2645 655 : goto END;
2646 : }
2647 36542 : IZ = pr_get_p(I);
2648 36542 : J = pr_inv_p(I);
2649 36542 : I = idealhnf_two(nf,I);
2650 36542 : break;
2651 101366 : case id_MAT:
2652 101366 : if (lg(I)-1 != N) pari_err_DIM("idealred");
2653 101359 : I = Q_primitive_part(I, &c1);
2654 101359 : IZ = gcoeff(I,1,1);
2655 101359 : if (is_pm1(IZ))
2656 : {
2657 8831 : if (!aI) { set_avma(av); return matid(N); }
2658 8747 : goto END;
2659 : }
2660 92528 : J = idealHNF_inv_Z(nf, I);
2661 92529 : break;
2662 21 : default: /* id_PRINCIPAL, silly case */
2663 21 : if (gequal0(I)) I = cgetg(1,t_MAT); else { c1 = I; I = matid(N); }
2664 21 : if (!aI) return I;
2665 14 : goto END;
2666 : }
2667 : /* now I integral, HNF; and J = (I\cap Z) I^(-1), integral */
2668 129071 : y = idealpseudomin(J, G); /* small elt in (I\cap Z)I^(-1), integral */
2669 129070 : if (ZV_isscalar(y))
2670 : { /* already reduced */
2671 71154 : if (!aI) return gerepilecopy(av, I);
2672 67975 : goto END;
2673 : }
2674 :
2675 57916 : my = zk_multable(nf, y);
2676 57915 : I = ZM_Z_divexact(ZM_mul(my, I), IZ); /* y I / (I\cap Z), integral */
2677 57917 : c1 = mul_content(c1, IZ);
2678 57917 : if (equali1(c1)) c1 = NULL; /* can be simplified with IZ */
2679 57917 : yi = ZM_gauss(my, col_ei(N,1)); /* y^-1 */
2680 57917 : dyi = Q_denom(yi); /* generates (y) \cap Z */
2681 57915 : I = hnfmodid(I, dyi);
2682 57917 : if (!aI) return gerepileupto(av, I);
2683 55944 : if (typ(aI) == t_MAT)
2684 : {
2685 39207 : GEN nyi = Q_muli_to_int(yi, dyi);
2686 39207 : if (gexpo(nyi) >= gexpo(y))
2687 20762 : aI = famat_div(aI, y); /* yi "larger" than y, keep the latter */
2688 : else
2689 : { /* use yi */
2690 18445 : aI = famat_mul(aI, nyi);
2691 18445 : c1 = div_content(c1, dyi);
2692 : }
2693 39207 : if (c1) { aI = famat_mul(aI, Q_to_famat(c1)); c1 = NULL; }
2694 : }
2695 : else
2696 16737 : c1 = c1? RgC_Rg_mul(yi, c1): yi;
2697 133335 : END:
2698 133335 : if (c1) aI = ext_mul(nf, aI,c1);
2699 133334 : return gerepilecopy(av, mkvec2(I, aI));
2700 : }
2701 :
2702 : /* I integral ZM (not HNF), G ZM, rounded Cholesky form of a weighted
2703 : * T2 matrix. Reduce I wrt G */
2704 : GEN
2705 1284125 : idealpseudored(GEN I, GEN G)
2706 1284125 : { return ZM_mul(I, ZM_lll(ZM_mul(G, I), 0.99, LLL_IM)); }
2707 :
2708 : /* Same I, G; m in I with T2(m) small */
2709 : GEN
2710 139964 : idealpseudomin(GEN I, GEN G)
2711 : {
2712 139964 : GEN u = ZM_lll(ZM_mul(G, I), 0.99, LLL_IM);
2713 139963 : return ZM_ZC_mul(I, gel(u,1));
2714 : }
2715 : /* Same I,G; irrational m in I with T2(m) small */
2716 : GEN
2717 0 : idealpseudomin_nonscalar(GEN I, GEN G)
2718 : {
2719 0 : GEN u = ZM_lll(ZM_mul(G, I), 0.99, LLL_IM);
2720 0 : GEN m = ZM_ZC_mul(I, gel(u,1));
2721 0 : if (ZV_isscalar(m) && lg(u) > 2) m = ZM_ZC_mul(I, gel(u,2));
2722 0 : return m;
2723 : }
2724 : /* Same I,G; t_VEC of irrational m in I with T2(m) small */
2725 : GEN
2726 1204391 : idealpseudominvec(GEN I, GEN G)
2727 : {
2728 1204391 : long i, j, k, n = lg(I)-1;
2729 1204391 : GEN x, L, b = idealpseudored(I, G);
2730 1204390 : L = cgetg(1 + (n*(n+1))/2, t_VEC);
2731 4256937 : for (i = k = 1; i <= n; i++)
2732 : {
2733 3052545 : x = gel(b,i);
2734 3052545 : if (!ZV_isscalar(x)) gel(L,k++) = x;
2735 : }
2736 3052547 : for (i = 2; i <= n; i++)
2737 : {
2738 1848157 : long J = minss(i, 4);
2739 4590738 : for (j = 1; j < J; j++)
2740 : {
2741 2742583 : x = ZC_add(gel(b,i),gel(b,j));
2742 2742581 : if (!ZV_isscalar(x)) gel(L,k++) = x;
2743 : }
2744 : }
2745 1204390 : setlg(L,k); return L;
2746 : }
2747 :
2748 : GEN
2749 10886 : idealred_elt(GEN nf, GEN I)
2750 : {
2751 10886 : pari_sp av = avma;
2752 10886 : GEN u = idealpseudomin(I, nf_get_roundG(nf));
2753 10886 : return gerepileupto(av, u);
2754 : }
2755 :
2756 : GEN
2757 7 : idealmin(GEN nf, GEN x, GEN vdir)
2758 : {
2759 7 : pari_sp av = avma;
2760 : GEN y, dx;
2761 7 : nf = checknf(nf);
2762 7 : switch( idealtyp(&x, NULL) )
2763 : {
2764 0 : case id_PRINCIPAL: return gcopy(x);
2765 0 : case id_PRIME: x = pr_hnf(nf,x); break;
2766 7 : case id_MAT: if (lg(x) == 1) return gen_0;
2767 : }
2768 7 : x = Q_remove_denom(x, &dx);
2769 7 : y = idealpseudomin(x, nf_get_Gtwist(nf,vdir));
2770 7 : if (dx) y = RgC_Rg_div(y, dx);
2771 7 : return gerepileupto(av, y);
2772 : }
2773 :
2774 : /*******************************************************************/
2775 : /* */
2776 : /* APPROXIMATION THEOREM */
2777 : /* */
2778 : /*******************************************************************/
2779 : /* a = ppi(a,b) ppo(a,b), where ppi regroups primes common to a and b
2780 : * and ppo(a,b) = Z_ppo(a,b) */
2781 : /* return gcd(a,b),ppi(a,b),ppo(a,b) */
2782 : GEN
2783 532721 : Z_ppio(GEN a, GEN b)
2784 : {
2785 532721 : GEN x, y, d = gcdii(a,b);
2786 532721 : if (is_pm1(d)) return mkvec3(gen_1, gen_1, a);
2787 413014 : x = d; y = diviiexact(a,d);
2788 : for(;;)
2789 68425 : {
2790 481439 : GEN g = gcdii(x,y);
2791 481439 : if (is_pm1(g)) return mkvec3(d, x, y);
2792 68425 : x = mulii(x,g); y = diviiexact(y,g);
2793 : }
2794 : }
2795 : /* a = ppg(a,b)pple(a,b), where ppg regroups primes such that v(a) > v(b)
2796 : * and pple all others */
2797 : /* return gcd(a,b),ppg(a,b),pple(a,b) */
2798 : GEN
2799 0 : Z_ppgle(GEN a, GEN b)
2800 : {
2801 0 : GEN x, y, g, d = gcdii(a,b);
2802 0 : if (equalii(a, d)) return mkvec3(a, gen_1, a);
2803 0 : x = diviiexact(a,d); y = d;
2804 : for(;;)
2805 : {
2806 0 : g = gcdii(x,y);
2807 0 : if (is_pm1(g)) return mkvec3(d, x, y);
2808 0 : x = mulii(x,g); y = diviiexact(y,g);
2809 : }
2810 : }
2811 : static void
2812 0 : Z_dcba_rec(GEN L, GEN a, GEN b)
2813 : {
2814 : GEN x, r, v, g, h, c, c0;
2815 : long n;
2816 0 : if (is_pm1(b)) {
2817 0 : if (!is_pm1(a)) vectrunc_append(L, a);
2818 0 : return;
2819 : }
2820 0 : v = Z_ppio(a,b);
2821 0 : a = gel(v,2);
2822 0 : r = gel(v,3);
2823 0 : if (!is_pm1(r)) vectrunc_append(L, r);
2824 0 : v = Z_ppgle(a,b);
2825 0 : g = gel(v,1);
2826 0 : h = gel(v,2);
2827 0 : x = c0 = gel(v,3);
2828 0 : for (n = 1; !is_pm1(h); n++)
2829 : {
2830 : GEN d, y;
2831 : long i;
2832 0 : v = Z_ppgle(h,sqri(g));
2833 0 : g = gel(v,1);
2834 0 : h = gel(v,2);
2835 0 : c = gel(v,3); if (is_pm1(c)) continue;
2836 0 : d = gcdii(c,b);
2837 0 : x = mulii(x,d);
2838 0 : y = d; for (i=1; i < n; i++) y = sqri(y);
2839 0 : Z_dcba_rec(L, diviiexact(c,y), d);
2840 : }
2841 0 : Z_dcba_rec(L,diviiexact(b,x), c0);
2842 : }
2843 : static GEN
2844 3764992 : Z_cba_rec(GEN L, GEN a, GEN b)
2845 : {
2846 : GEN g;
2847 : /* a few naive steps before switching to dcba */
2848 3764992 : if (lg(L) > 10) { Z_dcba_rec(L, a, b); return veclast(L); }
2849 3764992 : if (is_pm1(a)) return b;
2850 2242877 : g = gcdii(a,b);
2851 2242877 : if (is_pm1(g)) { vectrunc_append(L, a); return b; }
2852 1676087 : a = diviiexact(a,g);
2853 1676087 : b = diviiexact(b,g);
2854 1676087 : return Z_cba_rec(L, Z_cba_rec(L, a, g), b);
2855 : }
2856 : GEN
2857 412818 : Z_cba(GEN a, GEN b)
2858 : {
2859 412818 : GEN L = vectrunc_init(expi(a) + expi(b) + 2);
2860 412818 : GEN t = Z_cba_rec(L, a, b);
2861 412818 : if (!is_pm1(t)) vectrunc_append(L, t);
2862 412818 : return L;
2863 : }
2864 : /* P = coprime base, extend it by b; TODO: quadratic for now */
2865 : GEN
2866 35 : ZV_cba_extend(GEN P, GEN b)
2867 : {
2868 35 : long i, l = lg(P);
2869 35 : GEN w = cgetg(l+1, t_VEC);
2870 133 : for (i = 1; i < l; i++)
2871 : {
2872 98 : GEN v = Z_cba(gel(P,i), b);
2873 98 : long nv = lg(v)-1;
2874 98 : gel(w,i) = vecslice(v, 1, nv-1); /* those divide P[i] but not b */
2875 98 : b = gel(v,nv);
2876 : }
2877 35 : gel(w,l) = b; return shallowconcat1(w);
2878 : }
2879 : GEN
2880 28 : ZV_cba(GEN v)
2881 : {
2882 28 : long i, l = lg(v);
2883 : GEN P;
2884 28 : if (l <= 2) return v;
2885 14 : P = Z_cba(gel(v,1), gel(v,2));
2886 42 : for (i = 3; i < l; i++) P = ZV_cba_extend(P, gel(v,i));
2887 14 : return P;
2888 : }
2889 :
2890 : /* write x = x1 x2, x2 maximal s.t. (x2,f) = 1, return x2 */
2891 : GEN
2892 60217769 : Z_ppo(GEN x, GEN f)
2893 : {
2894 : for (;;)
2895 : {
2896 60217769 : f = gcdii(x, f); if (is_pm1(f)) break;
2897 40308259 : x = diviiexact(x, f);
2898 : }
2899 19907445 : return x;
2900 : }
2901 : /* write x = x1 x2, x2 maximal s.t. (x2,f) = 1, return x2 */
2902 : ulong
2903 69513879 : u_ppo(ulong x, ulong f)
2904 : {
2905 : for (;;)
2906 : {
2907 69513879 : f = ugcd(x, f); if (f == 1) break;
2908 15621925 : x /= f;
2909 : }
2910 53891925 : return x;
2911 : }
2912 :
2913 : /* result known to be representable as an ulong */
2914 : static ulong
2915 1508587 : lcmuu(ulong a, ulong b) { ulong d = ugcd(a,b); return (a/d) * b; }
2916 :
2917 : /* assume 0 < x < N; return u in (Z/NZ)^* such that u x = gcd(x,N) (mod N);
2918 : * set *pd = gcd(x,N) */
2919 : ulong
2920 5555992 : Fl_invgen(ulong x, ulong N, ulong *pd)
2921 : {
2922 : ulong d, d0, e, v, v1;
2923 : long s;
2924 5555992 : *pd = d = xgcduu(N, x, 0, &v, &v1, &s);
2925 5556628 : if (s > 0) v = N - v;
2926 5556628 : if (d == 1) return v;
2927 : /* vx = gcd(x,N) (mod N), v coprime to N/d but need not be coprime to N */
2928 2593165 : e = N / d;
2929 2593165 : d0 = u_ppo(d, e); /* d = d0 d1, d0 coprime to N/d, rad(d1) | N/d */
2930 2593295 : if (d0 == 1) return v;
2931 1508530 : e = lcmuu(e, d / d0);
2932 1508579 : return u_chinese_coprime(v, 1, e, d0, e*d0);
2933 : }
2934 :
2935 : /* x t_INT, f ideal. Write x = x1 x2, sqf(x1) | f, (x2,f) = 1. Return x2 */
2936 : static GEN
2937 126 : nf_coprime_part(GEN nf, GEN x, GEN listpr)
2938 : {
2939 126 : long v, j, lp = lg(listpr), N = nf_get_degree(nf);
2940 : GEN x1, x2, ex;
2941 :
2942 : #if 0 /*1) via many gcds. Expensive ! */
2943 : GEN f = idealprodprime(nf, listpr);
2944 : f = ZM_hnfmodid(f, x); /* first gcd is less expensive since x in Z */
2945 : x = scalarmat(x, N);
2946 : for (;;)
2947 : {
2948 : if (gequal1(gcoeff(f,1,1))) break;
2949 : x = idealdivexact(nf, x, f);
2950 : f = ZM_hnfmodid(shallowconcat(f,x), gcoeff(x,1,1)); /* gcd(f,x) */
2951 : }
2952 : x2 = x;
2953 : #else /*2) from prime decomposition */
2954 126 : x1 = NULL;
2955 350 : for (j=1; j<lp; j++)
2956 : {
2957 224 : GEN pr = gel(listpr,j);
2958 224 : v = Z_pval(x, pr_get_p(pr)); if (!v) continue;
2959 :
2960 126 : ex = muluu(v, pr_get_e(pr)); /* = v_pr(x) > 0 */
2961 126 : x1 = x1? idealmulpowprime(nf, x1, pr, ex)
2962 126 : : idealpow(nf, pr, ex);
2963 : }
2964 126 : x = scalarmat(x, N);
2965 126 : x2 = x1? idealdivexact(nf, x, x1): x;
2966 : #endif
2967 126 : return x2;
2968 : }
2969 :
2970 : /* L0 in K^*, assume (L0,f) = 1. Return L integral, L0 = L mod f */
2971 : GEN
2972 10920 : make_integral(GEN nf, GEN L0, GEN f, GEN listpr)
2973 : {
2974 : GEN fZ, t, L, D2, d1, d2, d;
2975 :
2976 10920 : L = Q_remove_denom(L0, &d);
2977 10920 : if (!d) return L0;
2978 :
2979 : /* L0 = L / d, L integral */
2980 518 : fZ = gcoeff(f,1,1);
2981 518 : if (typ(L) == t_INT) return Fp_mul(L, Fp_inv(d, fZ), fZ);
2982 : /* Kill denom part coprime to fZ */
2983 126 : d2 = Z_ppo(d, fZ);
2984 126 : t = Fp_inv(d2, fZ); if (!is_pm1(t)) L = ZC_Z_mul(L,t);
2985 126 : if (equalii(d, d2)) return L;
2986 :
2987 126 : d1 = diviiexact(d, d2);
2988 : /* L0 = (L / d1) mod f. d1 not coprime to f
2989 : * write (d1) = D1 D2, D2 minimal, (D2,f) = 1. */
2990 126 : D2 = nf_coprime_part(nf, d1, listpr);
2991 126 : t = idealaddtoone_i(nf, D2, f); /* in D2, 1 mod f */
2992 126 : L = nfmuli(nf,t,L);
2993 :
2994 : /* if (L0, f) = 1, then L in D1 ==> in D1 D2 = (d1) */
2995 126 : return Q_div_to_int(L, d1); /* exact division */
2996 : }
2997 :
2998 : /* assume L is a list of prime ideals. Return the product */
2999 : GEN
3000 336 : idealprodprime(GEN nf, GEN L)
3001 : {
3002 336 : long l = lg(L), i;
3003 : GEN z;
3004 336 : if (l == 1) return matid(nf_get_degree(nf));
3005 336 : z = pr_hnf(nf, gel(L,1));
3006 364 : for (i=2; i<l; i++) z = idealHNF_mul_two(nf,z, gel(L,i));
3007 336 : return z;
3008 : }
3009 :
3010 : /* optimize for the frequent case I = nfhnf()[2]: lots of them are 1 */
3011 : GEN
3012 462 : idealprod(GEN nf, GEN I)
3013 : {
3014 462 : long i, l = lg(I);
3015 : GEN z;
3016 1134 : for (i = 1; i < l; i++)
3017 1127 : if (!equali1(gel(I,i))) break;
3018 462 : if (i == l) return gen_1;
3019 455 : z = gel(I,i);
3020 763 : for (i++; i<l; i++) z = idealmul(nf, z, gel(I,i));
3021 455 : return z;
3022 : }
3023 :
3024 : /* v_pr(idealprod(nf,I)) */
3025 : long
3026 2806 : idealprodval(GEN nf, GEN I, GEN pr)
3027 : {
3028 2806 : long i, l = lg(I), v = 0;
3029 16006 : for (i = 1; i < l; i++)
3030 13200 : if (!equali1(gel(I,i))) v += idealval(nf, gel(I,i), pr);
3031 2806 : return v;
3032 : }
3033 :
3034 : /* assume L is a list of prime ideals. Return prod L[i]^e[i] */
3035 : GEN
3036 57565 : factorbackprime(GEN nf, GEN L, GEN e)
3037 : {
3038 57565 : long l = lg(L), i;
3039 : GEN z;
3040 :
3041 57565 : if (l == 1) return matid(nf_get_degree(nf));
3042 44153 : z = idealpow(nf, gel(L,1), gel(e,1));
3043 73770 : for (i=2; i<l; i++)
3044 29616 : if (signe(gel(e,i))) z = idealmulpowprime(nf,z, gel(L,i),gel(e,i));
3045 44154 : return z;
3046 : }
3047 :
3048 : /* F in Z, divisible exactly by pr.p. Return F-uniformizer for pr, i.e.
3049 : * a t in Z_K such that v_pr(t) = 1 and (t, F/pr) = 1 */
3050 : GEN
3051 57953 : pr_uniformizer(GEN pr, GEN F)
3052 : {
3053 57953 : GEN p = pr_get_p(pr), t = pr_get_gen(pr);
3054 57953 : if (!equalii(F, p))
3055 : {
3056 36616 : long e = pr_get_e(pr);
3057 36616 : GEN u, v, q = (e == 1)? sqri(p): p;
3058 36616 : u = mulii(q, Fp_inv(q, diviiexact(F,p))); /* 1 mod F/p, 0 mod q */
3059 36616 : v = subui(1UL, u); /* 0 mod F/p, 1 mod q */
3060 36616 : if (pr_is_inert(pr))
3061 28 : t = addii(mulii(p, v), u);
3062 : else
3063 : {
3064 36588 : t = ZC_Z_mul(t, v);
3065 36588 : gel(t,1) = addii(gel(t,1), u); /* return u + vt */
3066 : }
3067 : }
3068 57952 : return t;
3069 : }
3070 : /* L = list of prime ideals, return lcm_i (L[i] \cap \ZM) */
3071 : GEN
3072 81847 : prV_lcm_capZ(GEN L)
3073 : {
3074 81847 : long i, r = lg(L);
3075 : GEN F;
3076 81847 : if (r == 1) return gen_1;
3077 69297 : F = pr_get_p(gel(L,1));
3078 122746 : for (i = 2; i < r; i++)
3079 : {
3080 53449 : GEN pr = gel(L,i), p = pr_get_p(pr);
3081 53449 : if (!dvdii(F, p)) F = mulii(F,p);
3082 : }
3083 69297 : return F;
3084 : }
3085 : /* v vector of prid. Return underlying list of rational primes */
3086 : GEN
3087 60501 : prV_primes(GEN v)
3088 : {
3089 60501 : long i, l = lg(v);
3090 60501 : GEN w = cgetg(l,t_VEC);
3091 196000 : for (i=1; i<l; i++) gel(w,i) = pr_get_p(gel(v,i));
3092 60501 : return ZV_sort_uniq(w);
3093 : }
3094 :
3095 : /* Given a prime ideal factorization with possibly zero or negative
3096 : * exponents, gives b such that v_p(b) = v_p(x) for all prime ideals pr | x
3097 : * and v_pr(b) >= 0 for all other pr.
3098 : * For optimal performance, all [anti-]uniformizers should be precomputed,
3099 : * but no support for this yet. If nored, do not reduce result. */
3100 : static GEN
3101 53889 : idealapprfact_i(GEN nf, GEN x, int nored)
3102 : {
3103 53889 : GEN d = NULL, z, L, e, e2, F;
3104 : long i, r;
3105 53889 : int hasden = 0;
3106 :
3107 53889 : nf = checknf(nf);
3108 53889 : L = gel(x,1);
3109 53889 : e = gel(x,2);
3110 53889 : F = prV_lcm_capZ(L);
3111 53889 : z = NULL; r = lg(e);
3112 136231 : for (i = 1; i < r; i++)
3113 : {
3114 82342 : long s = signe(gel(e,i));
3115 : GEN pi, q;
3116 82342 : if (!s) continue;
3117 53585 : if (s < 0) hasden = 1;
3118 53585 : pi = pr_uniformizer(gel(L,i), F);
3119 53584 : q = nfpow(nf, pi, gel(e,i));
3120 53585 : z = z? nfmul(nf, z, q): q;
3121 : }
3122 53889 : if (!z) return gen_1;
3123 26785 : if (hasden) /* denominator */
3124 : {
3125 10107 : z = Q_remove_denom(z, &d);
3126 10107 : d = diviiexact(d, Z_ppo(d, F));
3127 : }
3128 26785 : if (nored || typ(z) != t_COL) return d? gdiv(z, d): z;
3129 10107 : e2 = cgetg(r, t_VEC);
3130 28679 : for (i = 1; i < r; i++) gel(e2,i) = addiu(gel(e,i), 1);
3131 10106 : x = factorbackprime(nf, L, e2);
3132 10107 : if (d) x = RgM_Rg_mul(x, d);
3133 10107 : z = ZC_reducemodlll(z, x);
3134 10107 : return d? RgC_Rg_div(z,d): z;
3135 : }
3136 :
3137 : GEN
3138 0 : idealapprfact(GEN nf, GEN x) {
3139 0 : pari_sp av = avma;
3140 0 : return gerepileupto(av, idealapprfact_i(nf, x, 0));
3141 : }
3142 : GEN
3143 14 : idealappr(GEN nf, GEN x) {
3144 14 : pari_sp av = avma;
3145 14 : if (!is_nf_extfactor(x)) x = idealfactor(nf, x);
3146 14 : return gerepileupto(av, idealapprfact_i(nf, x, 0));
3147 : }
3148 :
3149 : /* OBSOLETE */
3150 : GEN
3151 14 : idealappr0(GEN nf, GEN x, long fl) { (void)fl; return idealappr(nf, x); }
3152 :
3153 : static GEN
3154 21 : mat_ideal_two_elt2(GEN nf, GEN x, GEN a)
3155 : {
3156 21 : GEN F = idealfactor(nf,a), P = gel(F,1), E = gel(F,2);
3157 21 : long i, r = lg(E);
3158 84 : for (i=1; i<r; i++) gel(E,i) = stoi( idealval(nf,x,gel(P,i)) );
3159 21 : return idealapprfact_i(nf,F,1);
3160 : }
3161 :
3162 : static void
3163 14 : not_in_ideal(GEN a) {
3164 14 : pari_err_DOMAIN("idealtwoelt2","element mod ideal", "!=", gen_0, a);
3165 0 : }
3166 : /* x integral in HNF, a an 'nf' */
3167 : static int
3168 28 : in_ideal(GEN x, GEN a)
3169 : {
3170 28 : switch(typ(a))
3171 : {
3172 14 : case t_INT: return dvdii(a, gcoeff(x,1,1));
3173 7 : case t_COL: return RgV_is_ZV(a) && !!hnf_invimage(x, a);
3174 7 : default: return 0;
3175 : }
3176 : }
3177 :
3178 : /* Given an integral ideal x and a in x, gives a b such that
3179 : * x = aZ_K + bZ_K using the approximation theorem */
3180 : GEN
3181 42 : idealtwoelt2(GEN nf, GEN x, GEN a)
3182 : {
3183 42 : pari_sp av = avma;
3184 : GEN cx, b;
3185 :
3186 42 : nf = checknf(nf);
3187 42 : a = nf_to_scalar_or_basis(nf, a);
3188 42 : x = idealhnf_shallow(nf,x);
3189 42 : if (lg(x) == 1)
3190 : {
3191 14 : if (!isintzero(a)) not_in_ideal(a);
3192 7 : set_avma(av); return gen_0;
3193 : }
3194 28 : x = Q_primitive_part(x, &cx);
3195 28 : if (cx) a = gdiv(a, cx);
3196 28 : if (!in_ideal(x, a)) not_in_ideal(a);
3197 21 : b = mat_ideal_two_elt2(nf, x, a);
3198 21 : if (typ(b) == t_COL)
3199 : {
3200 14 : GEN mod = idealhnf_principal(nf,a);
3201 14 : b = ZC_hnfrem(b,mod);
3202 14 : if (ZV_isscalar(b)) b = gel(b,1);
3203 : }
3204 : else
3205 : {
3206 7 : GEN aZ = typ(a) == t_COL? Q_denom(zk_inv(nf,a)): a; /* (a) \cap Z */
3207 7 : b = centermodii(b, aZ, shifti(aZ,-1));
3208 : }
3209 21 : b = cx? gmul(b,cx): gcopy(b);
3210 21 : return gerepileupto(av, b);
3211 : }
3212 :
3213 : /* Given 2 integral ideals x and y in nf, returns a beta in nf such that
3214 : * beta * x is an integral ideal coprime to y */
3215 : GEN
3216 37204 : idealcoprimefact(GEN nf, GEN x, GEN fy)
3217 : {
3218 37204 : GEN L = gel(fy,1), e;
3219 37204 : long i, r = lg(L);
3220 :
3221 37204 : e = cgetg(r, t_COL);
3222 76068 : for (i=1; i<r; i++) gel(e,i) = stoi( -idealval(nf,x,gel(L,i)) );
3223 37204 : return idealapprfact_i(nf, mkmat2(L,e), 0);
3224 : }
3225 : GEN
3226 84 : idealcoprime(GEN nf, GEN x, GEN y)
3227 : {
3228 84 : pari_sp av = avma;
3229 84 : return gerepileupto(av, idealcoprimefact(nf, x, idealfactor(nf,y)));
3230 : }
3231 :
3232 : GEN
3233 7 : nfmulmodpr(GEN nf, GEN x, GEN y, GEN modpr)
3234 : {
3235 7 : pari_sp av = avma;
3236 7 : GEN z, p, pr = modpr, T;
3237 :
3238 7 : nf = checknf(nf); modpr = nf_to_Fq_init(nf,&pr,&T,&p);
3239 0 : x = nf_to_Fq(nf,x,modpr);
3240 0 : y = nf_to_Fq(nf,y,modpr);
3241 0 : z = Fq_mul(x,y,T,p);
3242 0 : return gerepileupto(av, algtobasis(nf, Fq_to_nf(z,modpr)));
3243 : }
3244 :
3245 : GEN
3246 0 : nfdivmodpr(GEN nf, GEN x, GEN y, GEN modpr)
3247 : {
3248 0 : pari_sp av = avma;
3249 0 : nf = checknf(nf);
3250 0 : return gerepileupto(av, nfreducemodpr(nf, nfdiv(nf,x,y), modpr));
3251 : }
3252 :
3253 : GEN
3254 0 : nfpowmodpr(GEN nf, GEN x, GEN k, GEN modpr)
3255 : {
3256 0 : pari_sp av=avma;
3257 0 : GEN z, T, p, pr = modpr;
3258 :
3259 0 : nf = checknf(nf); modpr = nf_to_Fq_init(nf,&pr,&T,&p);
3260 0 : z = nf_to_Fq(nf,x,modpr);
3261 0 : z = Fq_pow(z,k,T,p);
3262 0 : return gerepileupto(av, algtobasis(nf, Fq_to_nf(z,modpr)));
3263 : }
3264 :
3265 : GEN
3266 0 : nfkermodpr(GEN nf, GEN x, GEN modpr)
3267 : {
3268 0 : pari_sp av = avma;
3269 0 : GEN T, p, pr = modpr;
3270 :
3271 0 : nf = checknf(nf); modpr = nf_to_Fq_init(nf, &pr,&T,&p);
3272 0 : if (typ(x)!=t_MAT) pari_err_TYPE("nfkermodpr",x);
3273 0 : x = nfM_to_FqM(x, nf, modpr);
3274 0 : return gerepilecopy(av, FqM_to_nfM(FqM_ker(x,T,p), modpr));
3275 : }
3276 :
3277 : GEN
3278 0 : nfsolvemodpr(GEN nf, GEN a, GEN b, GEN pr)
3279 : {
3280 0 : const char *f = "nfsolvemodpr";
3281 0 : pari_sp av = avma;
3282 : GEN T, p, modpr;
3283 :
3284 0 : nf = checknf(nf);
3285 0 : modpr = nf_to_Fq_init(nf, &pr,&T,&p);
3286 0 : if (typ(a)!=t_MAT) pari_err_TYPE(f,a);
3287 0 : a = nfM_to_FqM(a, nf, modpr);
3288 0 : switch(typ(b))
3289 : {
3290 0 : case t_MAT:
3291 0 : b = nfM_to_FqM(b, nf, modpr);
3292 0 : b = FqM_gauss(a,b,T,p);
3293 0 : if (!b) pari_err_INV(f,a);
3294 0 : a = FqM_to_nfM(b, modpr);
3295 0 : break;
3296 0 : case t_COL:
3297 0 : b = nfV_to_FqV(b, nf, modpr);
3298 0 : b = FqM_FqC_gauss(a,b,T,p);
3299 0 : if (!b) pari_err_INV(f,a);
3300 0 : a = FqV_to_nfV(b, modpr);
3301 0 : break;
3302 0 : default: pari_err_TYPE(f,b);
3303 : }
3304 0 : return gerepilecopy(av, a);
3305 : }
|
