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 6736304 : idealtyp(GEN *ideal, GEN *arch)
47 : {
48 6736304 : GEN x = *ideal;
49 6736304 : long t,lx,tx = typ(x);
50 :
51 6736304 : if (tx!=t_VEC || lg(x)!=3) { if (arch) *arch = NULL; }
52 : else
53 : {
54 245768 : GEN a = gel(x,2);
55 245768 : 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 245754 : if (arch) *arch = a;
62 245761 : x = gel(x,1); tx = typ(x);
63 : }
64 6736297 : switch(tx)
65 : {
66 3815811 : case t_MAT: lx = lg(x);
67 3815811 : if (lx == 1) { t = id_PRINCIPAL; x = gen_0; break; }
68 3815650 : if (lx != lgcols(x)) pari_err_TYPE("idealtyp [nonsquare t_MAT]",x);
69 3815634 : t = id_MAT;
70 3815634 : break;
71 :
72 2154855 : case t_VEC: if (lg(x)!=6) pari_err_TYPE("idealtyp",x);
73 2154820 : t = id_PRIME; break;
74 :
75 765779 : case t_POL: case t_POLMOD: case t_COL:
76 : case t_INT: case t_FRAC:
77 765779 : t = id_PRINCIPAL; break;
78 0 : default:
79 0 : pari_err_TYPE("idealtyp",x);
80 : return 0; /*LCOV_EXCL_LINE*/
81 : }
82 6736394 : *ideal = x; return t;
83 : }
84 :
85 : /* true nf; v = [a,x,...], a in Z. Return (a,x) */
86 : GEN
87 642774 : idealhnf_two(GEN nf, GEN v)
88 : {
89 642774 : GEN p = gel(v,1), pi = gel(v,2), m = zk_scalar_or_multable(nf, pi);
90 642779 : if (typ(m) == t_INT) return scalarmat(gcdii(m,p), nf_get_degree(nf));
91 573427 : return ZM_hnfmodid(m, p);
92 : }
93 : /* true nf */
94 : GEN
95 2906462 : pr_hnf(GEN nf, GEN pr)
96 : {
97 2906462 : GEN p = pr_get_p(pr), m;
98 2906422 : if (pr_is_inert(pr)) return scalarmat(p, nf_get_degree(nf));
99 2479856 : m = zk_scalar_or_multable(nf, pr_get_gen(pr));
100 2479572 : return ZM_hnfmodprime(m, p);
101 : }
102 :
103 : GEN
104 682126 : idealhnf_principal(GEN nf, GEN x)
105 : {
106 : GEN cx;
107 682126 : x = nf_to_scalar_or_basis(nf, x);
108 682127 : switch(typ(x))
109 : {
110 434424 : case t_COL: break;
111 216777 : case t_INT: if (!signe(x)) return cgetg(1,t_MAT);
112 216014 : return scalarmat(absi_shallow(x), nf_get_degree(nf));
113 30926 : case t_FRAC:
114 30926 : return scalarmat(Q_abs_shallow(x), nf_get_degree(nf));
115 0 : default: pari_err_TYPE("idealhnf",x);
116 : }
117 434424 : x = Q_primitive_part(x, &cx);
118 434417 : RgV_check_ZV(x, "idealhnf");
119 434417 : x = zk_multable(nf, x);
120 434418 : x = ZM_hnfmodid(x, zkmultable_capZ(x));
121 434420 : return cx? ZM_Q_mul(x,cx): x;
122 : }
123 :
124 : /* x integral ideal in t_MAT form, nx columns */
125 : static GEN
126 7 : vec_mulid(GEN nf, GEN x, long nx, long N)
127 : {
128 7 : GEN m = cgetg(nx*N + 1, t_MAT);
129 : long i, j, k;
130 21 : for (i=k=1; i<=nx; i++)
131 56 : for (j=1; j<=N; j++) gel(m, k++) = zk_ei_mul(nf, gel(x,i),j);
132 7 : return m;
133 : }
134 : /* true nf */
135 : GEN
136 953675 : idealhnf_shallow(GEN nf, GEN x)
137 : {
138 953675 : long tx = typ(x), lx = lg(x), N;
139 :
140 : /* cannot use idealtyp because here we allow nonsquare matrices */
141 953675 : if (tx == t_VEC && lx == 3) { x = gel(x,1); tx = typ(x); lx = lg(x); }
142 953675 : if (tx == t_VEC && lx == 6) return pr_hnf(nf,x); /* PRIME */
143 577785 : switch(tx)
144 : {
145 97536 : case t_MAT:
146 : {
147 : GEN cx;
148 97536 : long nx = lx-1;
149 97536 : N = nf_get_degree(nf);
150 97536 : if (nx == 0) return cgetg(1, t_MAT);
151 97515 : if (nbrows(x) != N) pari_err_TYPE("idealhnf [wrong dimension]",x);
152 97508 : if (nx == 1) return idealhnf_principal(nf, gel(x,1));
153 :
154 81555 : if (nx == N && RgM_is_ZM(x) && ZM_ishnf(x)) return x;
155 47426 : x = Q_primitive_part(x, &cx);
156 47426 : if (nx < N) x = vec_mulid(nf, x, nx, N);
157 47426 : x = ZM_hnfmod(x, ZM_detmult(x));
158 47426 : return cx? ZM_Q_mul(x,cx): x;
159 : }
160 14 : case t_QFB:
161 : {
162 14 : pari_sp av = avma;
163 14 : GEN u, D = nf_get_disc(nf), T = nf_get_pol(nf), f = nf_get_index(nf);
164 14 : GEN A = gel(x,1), B = gel(x,2);
165 14 : N = nf_get_degree(nf);
166 14 : if (N != 2)
167 0 : pari_err_TYPE("idealhnf [Qfb for nonquadratic fields]", x);
168 14 : if (!equalii(qfb_disc(x), D))
169 7 : pari_err_DOMAIN("idealhnf [Qfb]", "disc(q)", "!=", D, x);
170 : /* x -> A Z + (-B + sqrt(D)) / 2 Z
171 : K = Q[t]/T(t), t^2 + ut + v = 0, u^2 - 4v = Df^2
172 : => t = (-u + sqrt(D) f)/2
173 : => sqrt(D)/2 = (t + u/2)/f */
174 7 : u = gel(T,3);
175 7 : B = deg1pol_shallow(ginv(f),
176 : gsub(gdiv(u, shifti(f,1)), gdiv(B,gen_2)),
177 7 : varn(T));
178 7 : return gerepileupto(av, idealhnf_two(nf, mkvec2(A,B)));
179 : }
180 480235 : default: return idealhnf_principal(nf, x); /* PRINCIPAL */
181 : }
182 : }
183 : /* true nf */
184 : GEN
185 252 : idealhnf(GEN nf, GEN x)
186 : {
187 252 : pari_sp av = avma;
188 252 : GEN y = idealhnf_shallow(nf, x);
189 245 : return (avma == av)? gcopy(y): gerepileupto(av, y);
190 : }
191 :
192 : /* GP functions */
193 :
194 : GEN
195 63 : idealtwoelt0(GEN nf, GEN x, GEN a)
196 : {
197 63 : if (!a) return idealtwoelt(nf,x);
198 42 : return idealtwoelt2(nf,x,a);
199 : }
200 :
201 : GEN
202 84 : idealpow0(GEN nf, GEN x, GEN n, long flag)
203 : {
204 84 : if (flag) return idealpowred(nf,x,n);
205 77 : return idealpow(nf,x,n);
206 : }
207 :
208 : GEN
209 70 : idealmul0(GEN nf, GEN x, GEN y, long flag)
210 : {
211 70 : if (flag) return idealmulred(nf,x,y);
212 63 : return idealmul(nf,x,y);
213 : }
214 :
215 : GEN
216 56 : idealdiv0(GEN nf, GEN x, GEN y, long flag)
217 : {
218 56 : switch(flag)
219 : {
220 28 : case 0: return idealdiv(nf,x,y);
221 28 : case 1: return idealdivexact(nf,x,y);
222 0 : default: pari_err_FLAG("idealdiv");
223 : }
224 : return NULL; /* LCOV_EXCL_LINE */
225 : }
226 :
227 : GEN
228 70 : idealaddtoone0(GEN nf, GEN arg1, GEN arg2)
229 : {
230 70 : if (!arg2) return idealaddmultoone(nf,arg1);
231 35 : return idealaddtoone(nf,arg1,arg2);
232 : }
233 :
234 : /* b not a scalar */
235 : static GEN
236 35 : hnf_Z_ZC(GEN nf, GEN a, GEN b) { return hnfmodid(zk_multable(nf,b), a); }
237 : /* b not a scalar */
238 : static GEN
239 28 : hnf_Z_QC(GEN nf, GEN a, GEN b)
240 : {
241 : GEN db;
242 28 : b = Q_remove_denom(b, &db);
243 28 : if (db) a = mulii(a, db);
244 28 : b = hnf_Z_ZC(nf,a,b);
245 28 : return db? RgM_Rg_div(b, db): b;
246 : }
247 : /* b not a scalar (not point in trying to optimize for this case) */
248 : static GEN
249 35 : hnf_Q_QC(GEN nf, GEN a, GEN b)
250 : {
251 : GEN da, db;
252 35 : if (typ(a) == t_INT) return hnf_Z_QC(nf, a, b);
253 7 : da = gel(a,2);
254 7 : a = gel(a,1);
255 7 : b = Q_remove_denom(b, &db);
256 : /* write da = d*A, db = d*B, gcd(A,B) = 1
257 : * gcd(a/(d A), b/(d B)) = gcd(a B, A b) / A B d = gcd(a B, b) / A B d */
258 7 : if (db)
259 : {
260 7 : GEN d = gcdii(da,db);
261 7 : if (!is_pm1(d)) db = diviiexact(db,d); /* B */
262 7 : if (!is_pm1(db))
263 : {
264 7 : a = mulii(a, db); /* a B */
265 7 : da = mulii(da, db); /* A B d = lcm(denom(a),denom(b)) */
266 : }
267 : }
268 7 : return RgM_Rg_div(hnf_Z_ZC(nf,a,b), da);
269 : }
270 : static GEN
271 7 : hnf_QC_QC(GEN nf, GEN a, GEN b)
272 : {
273 : GEN da, db, d, x;
274 7 : a = Q_remove_denom(a, &da);
275 7 : b = Q_remove_denom(b, &db);
276 7 : if (da) b = ZC_Z_mul(b, da);
277 7 : if (db) a = ZC_Z_mul(a, db);
278 7 : d = mul_denom(da, db);
279 7 : a = zk_multable(nf,a); da = zkmultable_capZ(a);
280 7 : b = zk_multable(nf,b); db = zkmultable_capZ(b);
281 7 : x = ZM_hnfmodid(shallowconcat(a,b), gcdii(da,db));
282 7 : return d? RgM_Rg_div(x, d): x;
283 : }
284 : static GEN
285 21 : hnf_Q_Q(GEN nf, GEN a, GEN b) {return scalarmat(Q_gcd(a,b), nf_get_degree(nf));}
286 : GEN
287 168 : idealhnf0(GEN nf, GEN a, GEN b)
288 : {
289 : long ta, tb;
290 : pari_sp av;
291 : GEN x;
292 168 : nf = checknf(nf);
293 168 : if (!b) return idealhnf(nf,a);
294 :
295 : /* HNF of aZ_K+bZ_K */
296 70 : av = avma;
297 70 : a = nf_to_scalar_or_basis(nf,a); ta = typ(a);
298 70 : b = nf_to_scalar_or_basis(nf,b); tb = typ(b);
299 63 : if (ta == t_COL)
300 14 : x = (tb==t_COL)? hnf_QC_QC(nf, a,b): hnf_Q_QC(nf, b,a);
301 : else
302 49 : x = (tb==t_COL)? hnf_Q_QC(nf, a,b): hnf_Q_Q(nf, a,b);
303 63 : return gerepileupto(av, x);
304 : }
305 :
306 : /*******************************************************************/
307 : /* */
308 : /* TWO-ELEMENT FORM */
309 : /* */
310 : /*******************************************************************/
311 : static GEN idealapprfact_i(GEN nf, GEN x, int nored);
312 :
313 : static int
314 440609 : ok_elt(GEN x, GEN xZ, GEN y)
315 : {
316 440609 : pari_sp av = avma;
317 440609 : return gc_bool(av, ZM_equal(x, ZM_hnfmodid(y, xZ)));
318 : }
319 :
320 : static GEN
321 141349 : addmul_col(GEN a, long s, GEN b)
322 : {
323 : long i,l;
324 141349 : if (!s) return a? leafcopy(a): a;
325 141096 : if (!a) return gmulsg(s,b);
326 130719 : l = lg(a);
327 605921 : for (i=1; i<l; i++)
328 475213 : if (signe(gel(b,i))) gel(a,i) = addii(gel(a,i), mulsi(s, gel(b,i)));
329 130708 : return a;
330 : }
331 :
332 : /* a <-- a + s * b, all coeffs integers */
333 : static GEN
334 75665 : addmul_mat(GEN a, long s, GEN b)
335 : {
336 : long j,l;
337 : /* copy otherwise next call corrupts a */
338 75665 : if (!s) return a? RgM_shallowcopy(a): a;
339 70554 : if (!a) return gmulsg(s,b);
340 35122 : l = lg(a);
341 153798 : for (j=1; j<l; j++)
342 118678 : (void)addmul_col(gel(a,j), s, gel(b,j));
343 35120 : return a;
344 : }
345 :
346 : static GEN
347 256907 : get_random_a(GEN nf, GEN x, GEN xZ)
348 : {
349 : pari_sp av;
350 256907 : long i, lm, l = lg(x);
351 : GEN a, z, beta, mul;
352 :
353 256907 : beta= cgetg(l, t_VEC);
354 256907 : mul = cgetg(l, t_VEC); lm = 1; /* = lg(mul) */
355 : /* look for a in x such that a O/xZ = x O/xZ */
356 475081 : for (i = 2; i < l; i++)
357 : {
358 464704 : GEN xi = gel(x,i);
359 464704 : GEN t = FpM_red(zk_multable(nf,xi), xZ); /* ZM, cannot be a scalar */
360 464695 : if (gequal0(t)) continue;
361 405178 : if (ok_elt(x,xZ, t)) return xi;
362 158651 : gel(beta,lm) = xi;
363 : /* mul[i] = { canonical generators for x[i] O/xZ as Z-module } */
364 158651 : gel(mul,lm) = t; lm++;
365 : }
366 10377 : setlg(mul, lm);
367 10377 : setlg(beta,lm);
368 10377 : z = cgetg(lm, t_VECSMALL);
369 35635 : for(av = avma;; set_avma(av))
370 : {
371 111300 : for (a=NULL,i=1; i<lm; i++)
372 : {
373 75665 : long t = random_bits(4) - 7; /* in [-7,8] */
374 75665 : z[i] = t;
375 75665 : a = addmul_mat(a, t, gel(mul,i));
376 : }
377 : /* a = matrix (NOT HNF) of ideal generated by beta.z in O/xZ */
378 35635 : if (a && ok_elt(x,xZ, a)) break;
379 : }
380 33048 : for (a=NULL,i=1; i<lm; i++)
381 22671 : a = addmul_col(a, z[i], gel(beta,i));
382 10377 : return a;
383 : }
384 :
385 : /* x square matrix, assume it is HNF */
386 : static GEN
387 405410 : mat_ideal_two_elt(GEN nf, GEN x)
388 : {
389 : GEN y, a, cx, xZ;
390 405410 : long N = nf_get_degree(nf);
391 : pari_sp av, tetpil;
392 :
393 405410 : if (lg(x)-1 != N) pari_err_DIM("idealtwoelt");
394 405396 : if (N == 2) return mkvec2copy(gcoeff(x,1,1), gel(x,2));
395 :
396 280647 : y = cgetg(3,t_VEC); av = avma;
397 280647 : cx = Q_content(x);
398 280648 : xZ = gcoeff(x,1,1);
399 280648 : if (gequal(xZ, cx)) /* x = (cx) */
400 : {
401 4966 : gel(y,1) = cx;
402 4966 : gel(y,2) = gen_0; return y;
403 : }
404 275682 : if (equali1(cx)) cx = NULL;
405 : else
406 : {
407 1533 : x = Q_div_to_int(x, cx);
408 1533 : xZ = gcoeff(x,1,1);
409 : }
410 275682 : if (N < 6)
411 238578 : a = get_random_a(nf, x, xZ);
412 : else
413 : {
414 37104 : const long FB[] = { _evallg(15+1) | evaltyp(t_VECSMALL),
415 : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
416 : };
417 37104 : GEN P, E, a1 = Z_lsmoothen(xZ, (GEN)FB, &P, &E);
418 37104 : if (!a1) /* factors completely */
419 18775 : a = idealapprfact_i(nf, idealfactor(nf,x), 1);
420 18329 : else if (lg(P) == 1) /* no small factors */
421 11029 : a = get_random_a(nf, x, xZ);
422 : else /* general case */
423 : {
424 : GEN A0, A1, a0, u0, u1, v0, v1, pi0, pi1, t, u;
425 7300 : a0 = diviiexact(xZ, a1);
426 7300 : A0 = ZM_hnfmodid(x, a0); /* smooth part of x */
427 7300 : A1 = ZM_hnfmodid(x, a1); /* cofactor */
428 7300 : pi0 = idealapprfact_i(nf, idealfactor(nf,A0), 1);
429 7300 : pi1 = get_random_a(nf, A1, a1);
430 7300 : (void)bezout(a0, a1, &v0,&v1);
431 7300 : u0 = mulii(a0, v0);
432 7300 : u1 = mulii(a1, v1);
433 7300 : if (typ(pi0) != t_COL) t = addmulii(u0, pi0, u1);
434 : else
435 7300 : { t = ZC_Z_mul(pi0, u1); gel(t,1) = addii(gel(t,1), u0); }
436 7300 : u = ZC_Z_mul(pi1, u0); gel(u,1) = addii(gel(u,1), u1);
437 7300 : a = nfmuli(nf, centermod(u, xZ), centermod(t, xZ));
438 : }
439 : }
440 275681 : if (cx)
441 : {
442 1533 : a = centermod(a, xZ);
443 1533 : tetpil = avma;
444 1533 : if (typ(cx) == t_INT)
445 : {
446 70 : gel(y,1) = mulii(xZ, cx);
447 70 : gel(y,2) = ZC_Z_mul(a, cx);
448 : }
449 : else
450 : {
451 1463 : gel(y,1) = gmul(xZ, cx);
452 1463 : gel(y,2) = RgC_Rg_mul(a, cx);
453 : }
454 : }
455 : else
456 : {
457 274148 : tetpil = avma;
458 274148 : gel(y,1) = icopy(xZ);
459 274149 : gel(y,2) = centermod(a, xZ);
460 : }
461 275681 : gerepilecoeffssp(av,tetpil,y+1,2); return y;
462 : }
463 :
464 : /* Given an ideal x, returns [a,alpha] such that a is in Q,
465 : * x = a Z_K + alpha Z_K, alpha in K^*
466 : * a = 0 or alpha = 0 are possible, but do not try to determine whether
467 : * x is principal. */
468 : GEN
469 99224 : idealtwoelt(GEN nf, GEN x)
470 : {
471 : pari_sp av;
472 99224 : long tx = idealtyp(&x, NULL);
473 99218 : nf = checknf(nf);
474 99218 : if (tx == id_MAT) return mat_ideal_two_elt(nf,x);
475 1050 : if (tx == id_PRIME) return mkvec2copy(gel(x,1), gel(x,2));
476 : /* id_PRINCIPAL */
477 1043 : av = avma; x = nf_to_scalar_or_basis(nf, x);
478 1890 : return gerepilecopy(av, typ(x)==t_COL? mkvec2(gen_0,x):
479 938 : mkvec2(Q_abs_shallow(x),gen_0));
480 : }
481 :
482 : /*******************************************************************/
483 : /* */
484 : /* FACTORIZATION */
485 : /* */
486 : /*******************************************************************/
487 : /* x integral ideal in HNF, Zval = v_p(x \cap Z) > 0; return v_p(Nx) */
488 : static long
489 734383 : idealHNF_norm_pval(GEN x, GEN p, long Zval)
490 : {
491 734383 : long i, v = Zval, l = lg(x);
492 2387319 : for (i = 2; i < l; i++) v += Z_pval(gcoeff(x,i,i), p);
493 734381 : return v;
494 : }
495 :
496 : /* x integral in HNF, f0 = partial factorization of a multiple of
497 : * x[1,1] = x\cap Z */
498 : GEN
499 278860 : idealHNF_Z_factor_i(GEN x, GEN f0, GEN *pvN, GEN *pvZ)
500 : {
501 278860 : GEN P, E, vN, vZ, xZ = gcoeff(x,1,1), f = f0? f0: Z_factor(xZ);
502 : long i, l;
503 278868 : P = gel(f,1); l = lg(P);
504 278868 : E = gel(f,2);
505 278868 : *pvN = vN = cgetg(l, t_VECSMALL);
506 278873 : *pvZ = vZ = cgetg(l, t_VECSMALL);
507 712471 : for (i = 1; i < l; i++)
508 : {
509 433602 : GEN p = gel(P,i);
510 433602 : vZ[i] = f0? Z_pval(xZ, p): (long) itou(gel(E,i));
511 433602 : vN[i] = idealHNF_norm_pval(x,p, vZ[i]);
512 : }
513 278869 : return P;
514 : }
515 : /* return P, primes dividing Nx and xZ = x\cap Z, set v_p(Nx), v_p(xZ);
516 : * x integral in HNF */
517 : GEN
518 0 : idealHNF_Z_factor(GEN x, GEN *pvN, GEN *pvZ)
519 0 : { return idealHNF_Z_factor_i(x, NULL, pvN, pvZ); }
520 :
521 : /* v_P(A)*f(P) <= Nval [e.g. Nval = v_p(Norm A)], Zval = v_p(A \cap Z).
522 : * Return v_P(A) */
523 : static long
524 905054 : idealHNF_val(GEN A, GEN P, long Nval, long Zval)
525 : {
526 905054 : long f = pr_get_f(P), vmax, v, e, i, j, k, l;
527 : GEN mul, B, a, y, r, p, pk, cx, vals;
528 : pari_sp av;
529 :
530 905054 : if (Nval < f) return 0;
531 904753 : p = pr_get_p(P);
532 904755 : e = pr_get_e(P);
533 : /* v_P(A) <= max [ e * v_p(A \cap Z), floor[v_p(Nix) / f ] */
534 904757 : vmax = minss(Zval * e, Nval / f);
535 904754 : mul = pr_get_tau(P);
536 904751 : l = lg(mul);
537 904751 : B = cgetg(l,t_MAT);
538 : /* B[1] not needed: v_pr(A[1]) = v_pr(A \cap Z) is known already */
539 905116 : gel(B,1) = gen_0; /* dummy */
540 2816116 : for (j = 2; j < l; j++)
541 : {
542 2149105 : GEN x = gel(A,j);
543 2149105 : gel(B,j) = y = cgetg(l, t_COL);
544 18119298 : for (i = 1; i < l; i++)
545 : { /* compute a = (x.t0)_i, A in HNF ==> x[j+1..l-1] = 0 */
546 16208298 : a = mulii(gel(x,1), gcoeff(mul,i,1));
547 143982379 : for (k = 2; k <= j; k++) a = addii(a, mulii(gel(x,k), gcoeff(mul,i,k)));
548 : /* p | a ? */
549 16207845 : gel(y,i) = dvmdii(a,p,&r); if (signe(r)) return 0;
550 : }
551 : }
552 667011 : vals = cgetg(l, t_VECSMALL);
553 : /* vals[1] not needed */
554 2378580 : for (j = 2; j < l; j++)
555 : {
556 1711490 : gel(B,j) = Q_primitive_part(gel(B,j), &cx);
557 1711507 : vals[j] = cx? 1 + e * Q_pval(cx, p): 1;
558 : }
559 667090 : pk = powiu(p, ceildivuu(vmax, e));
560 667072 : av = avma; y = cgetg(l,t_COL);
561 : /* can compute mod p^ceil((vmax-v)/e) */
562 1262857 : for (v = 1; v < vmax; v++)
563 : { /* we know v_pr(Bj) >= v for all j */
564 622429 : if (e == 1 || (vmax - v) % e == 0) pk = diviiexact(pk, p);
565 3609366 : for (j = 2; j < l; j++)
566 : {
567 3013612 : GEN x = gel(B,j); if (v < vals[j]) continue;
568 18288305 : for (i = 1; i < l; i++)
569 : {
570 16552586 : pari_sp av2 = avma;
571 16552586 : a = mulii(gel(x,1), gcoeff(mul,i,1));
572 262829377 : for (k = 2; k < l; k++) a = addii(a, mulii(gel(x,k), gcoeff(mul,i,k)));
573 : /* a = (x.t_0)_i; p | a ? */
574 16551056 : a = dvmdii(a,p,&r); if (signe(r)) return v;
575 16524368 : if (lgefint(a) > lgefint(pk)) a = remii(a, pk);
576 16524393 : gel(y,i) = gerepileuptoint(av2, a);
577 : }
578 1735719 : gel(B,j) = y; y = x;
579 1735719 : if (gc_needed(av,3))
580 : {
581 0 : if(DEBUGMEM>1) pari_warn(warnmem,"idealval");
582 0 : gerepileall(av,3, &y,&B,&pk);
583 : }
584 : }
585 : }
586 640428 : return v;
587 : }
588 : /* true nf, x != 0 integral ideal in HNF, cx t_INT or NULL,
589 : * FA integer factorization matrix or NULL. Return partial factorization of
590 : * cx * x above primes in FA (complete factorization if !FA)*/
591 : static GEN
592 278856 : idealHNF_factor_i(GEN nf, GEN x, GEN cx, GEN FA)
593 : {
594 278856 : const long N = lg(x)-1;
595 : long i, j, k, l, v;
596 278856 : GEN vN, vZ, vP, vE, vp = idealHNF_Z_factor_i(x, FA, &vN,&vZ);
597 :
598 278868 : l = lg(vp);
599 278868 : i = cx? expi(cx)+1: 1;
600 278876 : vP = cgetg((l+i-2)*N+1, t_COL);
601 278869 : vE = cgetg((l+i-2)*N+1, t_COL);
602 712447 : for (i = k = 1; i < l; i++)
603 : {
604 433594 : GEN L, p = gel(vp,i);
605 433594 : long Nval = vN[i], Zval = vZ[i], vc = cx? Z_pvalrem(cx,p,&cx): 0;
606 433585 : if (vc)
607 : {
608 45616 : L = idealprimedec(nf,p);
609 45618 : if (is_pm1(cx)) cx = NULL;
610 : }
611 : else
612 387969 : L = idealprimedec_limit_f(nf,p,Nval);
613 1037834 : for (j = 1; Nval && j < lg(L); j++) /* !Nval => only cx contributes */
614 : {
615 604264 : GEN P = gel(L,j);
616 604264 : pari_sp av = avma;
617 604264 : v = idealHNF_val(x, P, Nval, Zval);
618 604231 : set_avma(av);
619 604233 : Nval -= v*pr_get_f(P);
620 604233 : v += vc * pr_get_e(P); if (!v) continue;
621 483940 : gel(vP,k) = P;
622 483940 : gel(vE,k) = utoipos(v); k++;
623 : }
624 477867 : if (vc) for (; j<lg(L); j++)
625 : {
626 44299 : GEN P = gel(L,j);
627 44299 : gel(vP,k) = P;
628 44299 : gel(vE,k) = utoipos(vc * pr_get_e(P)); k++;
629 : }
630 : }
631 278853 : if (cx && !FA)
632 : { /* complete factorization */
633 60086 : GEN f = Z_factor(cx), cP = gel(f,1), cE = gel(f,2);
634 60086 : long lc = lg(cP);
635 128070 : for (i=1; i<lc; i++)
636 : {
637 67982 : GEN p = gel(cP,i), L = idealprimedec(nf,p);
638 67984 : long vc = itos(gel(cE,i));
639 146454 : for (j=1; j<lg(L); j++)
640 : {
641 78470 : GEN P = gel(L,j);
642 78470 : gel(vP,k) = P;
643 78470 : gel(vE,k) = utoipos(vc * pr_get_e(P)); k++;
644 : }
645 : }
646 : }
647 278855 : setlg(vP, k);
648 278867 : setlg(vE, k); return mkmat2(vP, vE);
649 : }
650 : /* true nf, x integral ideal */
651 : static GEN
652 233558 : idealHNF_factor(GEN nf, GEN x, ulong lim)
653 : {
654 233558 : GEN cx, F = NULL;
655 233558 : if (lim)
656 : {
657 : GEN P, E;
658 : long i;
659 : /* strict useless because of prime table */
660 42 : F = absZ_factor_limit(gcoeff(x,1,1), lim);
661 42 : P = gel(F,1);
662 42 : E = gel(F,2);
663 : /* filter out entries > lim */
664 77 : for (i = lg(P)-1; i; i--)
665 77 : if (cmpiu(gel(P,i), lim) <= 0) break;
666 42 : setlg(P, i+1);
667 42 : setlg(E, i+1);
668 : }
669 233558 : x = Q_primitive_part(x, &cx);
670 233549 : return idealHNF_factor_i(nf, x, cx, F);
671 : }
672 : /* c * vector(#L,i,L[i].e), assume results fit in ulong */
673 : static GEN
674 33025 : prV_e_muls(GEN L, long c)
675 : {
676 33025 : long j, l = lg(L);
677 33025 : GEN z = cgetg(l, t_COL);
678 67574 : for (j = 1; j < l; j++) gel(z,j) = stoi(c * pr_get_e(gel(L,j)));
679 33034 : return z;
680 : }
681 : /* true nf, y in Q */
682 : static GEN
683 26317 : Q_nffactor(GEN nf, GEN y, ulong lim)
684 : {
685 : GEN f, P, E;
686 : long l, i;
687 26317 : if (typ(y) == t_INT)
688 : {
689 26294 : if (!signe(y)) pari_err_DOMAIN("idealfactor", "ideal", "=",gen_0,y);
690 26280 : if (is_pm1(y)) return trivial_fact();
691 : }
692 17249 : y = Q_abs_shallow(y);
693 17256 : if (!lim) f = Q_factor(y);
694 : else
695 : {
696 35 : f = Q_factor_limit(y, lim);
697 35 : P = gel(f,1);
698 35 : E = gel(f,2);
699 77 : for (i = lg(P)-1; i > 0; i--)
700 63 : if (abscmpiu(gel(P,i), lim) < 0) break;
701 35 : setlg(P,i+1); setlg(E,i+1);
702 : }
703 17247 : P = gel(f,1); l = lg(P); if (l == 1) return f;
704 17233 : E = gel(f,2);
705 50310 : for (i = 1; i < l; i++)
706 : {
707 33056 : gel(P,i) = idealprimedec(nf, gel(P,i));
708 33009 : gel(E,i) = prV_e_muls(gel(P,i), itos(gel(E,i)));
709 : }
710 17254 : P = shallowconcat1(P); gel(f,1) = P; settyp(P, t_COL);
711 17275 : E = shallowconcat1(E); gel(f,2) = E; return f;
712 : }
713 :
714 : GEN
715 3696 : idealfactor_partial(GEN nf, GEN x, GEN L)
716 : {
717 3696 : pari_sp av = avma;
718 : long i, j, l;
719 : GEN P, E;
720 3696 : if (!L) return idealfactor(nf, x);
721 2975 : if (typ(L) == t_INT) return idealfactor_limit(nf, x, itou(L));
722 2954 : l = lg(L); if (l == 1) return trivial_fact();
723 2170 : P = cgetg(l, t_VEC);
724 4480 : for (i = 1; i < l; i++)
725 : {
726 2310 : GEN p = gel(L,i);
727 2310 : gel(P,i) = typ(p) == t_INT? idealprimedec(nf, p): mkvec(p);
728 : }
729 2170 : P = shallowconcat1(P); settyp(P, t_COL);
730 2170 : P = gen_sort_uniq(P, (void*)&cmp_prime_ideal, &cmp_nodata);
731 2170 : E = cgetg_copy(P, &l);
732 12215 : for (i = j = 1; i < l; i++)
733 : {
734 10045 : long v = idealval(nf, x, gel(P,i));
735 10045 : if (v) { gel(P,j) = gel(P,i); gel(E,j) = stoi(v); j++; }
736 : }
737 2170 : setlg(P,j);
738 2170 : setlg(E,j); return gerepilecopy(av, mkmat2(P, E));
739 : }
740 : GEN
741 259961 : idealfactor_limit(GEN nf, GEN x, ulong lim)
742 : {
743 259961 : pari_sp av = avma;
744 : GEN fa, y;
745 259961 : long tx = idealtyp(&x, NULL);
746 :
747 259956 : if (tx == id_PRIME)
748 : {
749 98 : if (lim && abscmpiu(pr_get_p(x), lim) >= 0) return trivial_fact();
750 91 : retmkmat2(mkcolcopy(x), mkcol(gen_1));
751 : }
752 259858 : nf = checknf(nf);
753 259858 : if (tx == id_PRINCIPAL)
754 : {
755 27806 : y = nf_to_scalar_or_basis(nf, x);
756 27807 : if (typ(y) != t_COL) return gerepilecopy(av, Q_nffactor(nf, y, lim));
757 : }
758 233534 : y = idealnumden(nf, x);
759 233547 : fa = idealHNF_factor(nf, gel(y,1), lim);
760 233495 : if (!isint1(gel(y,2)))
761 14 : fa = famat_div_shallow(fa, idealHNF_factor(nf, gel(y,2), lim));
762 233547 : fa = gerepilecopy(av, fa);
763 233555 : return sort_factor(fa, (void*)&cmp_prime_ideal, &cmp_nodata);
764 : }
765 : GEN
766 259761 : idealfactor(GEN nf, GEN x) { return idealfactor_limit(nf, x, 0); }
767 : GEN
768 182 : gpidealfactor(GEN nf, GEN x, GEN lim)
769 : {
770 182 : ulong L = 0;
771 182 : if (lim)
772 : {
773 70 : if (typ(lim) != t_INT || signe(lim) < 0) pari_err_FLAG("idealfactor");
774 70 : L = itou(lim);
775 : }
776 182 : return idealfactor_limit(nf, x, L);
777 : }
778 :
779 : static GEN
780 7532 : ramified_root(GEN nf, GEN R, GEN A, long n)
781 : {
782 7532 : GEN v, P = gel(idealfactor(nf, R), 1);
783 7532 : long i, l = lg(P);
784 7532 : v = cgetg(l, t_VECSMALL);
785 7994 : for (i = 1; i < l; i++)
786 : {
787 469 : long w = idealval(nf, A, gel(P,i));
788 469 : if (w % n) return NULL;
789 462 : v[i] = w / n;
790 : }
791 7525 : return idealfactorback(nf, P, v, 0);
792 : }
793 : static int
794 0 : ramified_root_simple(GEN nf, long n, GEN P, GEN v)
795 : {
796 0 : long i, l = lg(v);
797 0 : for (i = 1; i < l; i++) if (v[i])
798 : {
799 0 : GEN vpr = idealprimedec(nf, gel(P,i));
800 0 : long lpr = lg(vpr), j;
801 0 : for (j = 1; j < lpr; j++)
802 : {
803 0 : long e = pr_get_e(gel(vpr,j));
804 0 : if ((e * v[i]) % n) return 0;
805 : }
806 : }
807 0 : return 1;
808 : }
809 : /* true nf; A is assumed to be the n-th power of an integral ideal,
810 : * return its n-th root; n > 1 */
811 : static long
812 7532 : idealsqrtn_int(GEN nf, GEN A, long n, GEN *pB)
813 : {
814 : GEN C, root;
815 : long i, l;
816 :
817 7532 : if (typ(A) == t_INT) /* > 0 */
818 : {
819 3766 : GEN P = nf_get_ramified_primes(nf), v, q;
820 3766 : l = lg(P); v = cgetg(l, t_VECSMALL);
821 17031 : for (i = 1; i < l; i++) v[i] = Z_pvalrem(A, gel(P,i), &A);
822 3766 : C = gen_1;
823 3766 : if (!isint1(A) && !Z_ispowerall(A, n, pB? &C: NULL)) return 0;
824 3766 : if (!pB) return ramified_root_simple(nf, n, P, v);
825 3766 : q = factorback2(P, v);
826 3766 : root = ramified_root(nf, q, q, n);
827 3766 : if (!root) return 0;
828 3766 : if (!equali1(C)) root = isint1(root)? C: ZM_Z_mul(root, C);
829 3766 : *pB = root; return 1;
830 : }
831 : /* compute valuations at ramified primes */
832 3766 : root = ramified_root(nf, idealadd(nf, nf_get_diff(nf), A), A, n);
833 3766 : if (!root) return 0;
834 : /* remove ramified primes */
835 3759 : if (isint1(root))
836 3325 : root = matid(nf_get_degree(nf));
837 : else
838 434 : A = idealdivexact(nf, A, idealpows(nf,root,n));
839 3759 : A = Q_primitive_part(A, &C);
840 3759 : if (C)
841 : {
842 245 : if (!Z_ispowerall(C,n,&C)) return 0;
843 245 : if (pB) root = ZM_Z_mul(root, C);
844 : }
845 :
846 : /* compute final n-th root, at most degree(nf)-1 iterations */
847 3759 : for (i = 0;; i++)
848 2387 : {
849 6146 : GEN J, b, a = gcoeff(A,1,1); /* A \cap Z */
850 6146 : if (is_pm1(a)) break;
851 2408 : if (!Z_ispowerall(a,n,&b)) return 0;
852 2387 : J = idealadd(nf, b, A);
853 2387 : A = idealdivexact(nf, idealpows(nf,J,n), A);
854 : /* div and not divexact here */
855 2387 : if (pB) root = odd(i)? idealdiv(nf, root, J): idealmul(nf, root, J);
856 : }
857 3738 : if (pB) *pB = root;
858 3738 : return 1;
859 : }
860 :
861 : /* A is assumed to be the n-th power of an ideal in nf
862 : returns its n-th root. */
863 : long
864 3787 : idealispower(GEN nf, GEN A, long n, GEN *pB)
865 : {
866 3787 : pari_sp av = avma;
867 : GEN v, N, D;
868 3787 : nf = checknf(nf);
869 3787 : if (n <= 0) pari_err_DOMAIN("idealispower", "n", "<=", gen_0, stoi(n));
870 3787 : if (n == 1) { if (pB) *pB = idealhnf(nf,A); return 1; }
871 3780 : v = idealnumden(nf,A);
872 3780 : if (gequal0(gel(v,1))) { set_avma(av); if (pB) *pB = cgetg(1,t_MAT); return 1; }
873 3780 : if (!idealsqrtn_int(nf, gel(v,1), n, pB? &N: NULL)) return 0;
874 3752 : if (!idealsqrtn_int(nf, gel(v,2), n, pB? &D: NULL)) return 0;
875 3752 : if (pB) *pB = gerepileupto(av, idealdiv(nf,N,D)); else set_avma(av);
876 3752 : return 1;
877 : }
878 :
879 : /* x t_INT or integral nonzero ideal in HNF */
880 : static GEN
881 96572 : idealredmodpower_i(GEN nf, GEN x, ulong k, ulong B)
882 : {
883 : GEN cx, y, U, N, F, Q;
884 96572 : if (typ(x) == t_INT)
885 : {
886 50687 : if (!signe(x) || is_pm1(x)) return gen_1;
887 1785 : F = Z_factor_limit(x, B);
888 1785 : gel(F,2) = gdiventgs(gel(F,2), k);
889 1785 : return ginv(factorback(F));
890 : }
891 45885 : N = gcoeff(x,1,1); if (is_pm1(N)) return gen_1;
892 45311 : F = absZ_factor_limit_strict(N, B, &U);
893 45310 : if (U)
894 : {
895 336 : GEN M = powii(gel(U,1), gel(U,2));
896 336 : y = hnfmodid(x, M); /* coprime part to B! */
897 336 : if (!idealispower(nf, y, k, &U)) U = NULL;
898 336 : x = hnfmodid(x, diviiexact(N, M));
899 : }
900 : /* x = B-smooth part of initial x */
901 45310 : x = Q_primitive_part(x, &cx);
902 45310 : F = idealHNF_factor_i(nf, x, cx, F);
903 45310 : gel(F,2) = gdiventgs(gel(F,2), k);
904 45311 : Q = idealfactorback(nf, gel(F,1), gel(F,2), 0);
905 45310 : if (U) Q = idealmul(nf,Q,U);
906 45311 : if (typ(Q) == t_INT) return Q;
907 13733 : y = idealred_elt(nf, idealHNF_inv_Z(nf, Q));
908 13733 : return gdiv(y, gcoeff(Q,1,1));
909 : }
910 : GEN
911 48293 : idealredmodpower(GEN nf, GEN x, ulong n, ulong B)
912 : {
913 48293 : pari_sp av = avma;
914 : GEN a, b;
915 48293 : nf = checknf(nf);
916 48293 : if (!n) pari_err_DOMAIN("idealredmodpower","n", "=", gen_0, gen_0);
917 48293 : x = idealnumden(nf, x);
918 48293 : a = gel(x,1);
919 48293 : if (isintzero(a)) { set_avma(av); return gen_1; }
920 48286 : a = idealredmodpower_i(nf, gel(x,1), n, B);
921 48286 : b = idealredmodpower_i(nf, gel(x,2), n, B);
922 48286 : if (!isint1(b)) a = nf_to_scalar_or_basis(nf, nfdiv(nf, a, b));
923 48286 : return gerepilecopy(av, a);
924 : }
925 :
926 : /* P prime ideal in idealprimedec format. Return valuation(A) at P */
927 : long
928 931474 : idealval(GEN nf, GEN A, GEN P)
929 : {
930 931474 : pari_sp av = avma;
931 : GEN p, cA;
932 931474 : long vcA, v, Zval, tx = idealtyp(&A, NULL);
933 :
934 931475 : if (tx == id_PRINCIPAL) return nfval(nf,A,P);
935 847704 : checkprid(P);
936 847705 : if (tx == id_PRIME) return pr_equal(P, A)? 1: 0;
937 : /* id_MAT */
938 847677 : nf = checknf(nf);
939 847677 : A = Q_primitive_part(A, &cA);
940 847676 : p = pr_get_p(P);
941 847677 : vcA = cA? Q_pval(cA,p): 0;
942 847677 : if (pr_is_inert(P)) return gc_long(av,vcA);
943 834798 : Zval = Z_pval(gcoeff(A,1,1), p);
944 834799 : if (!Zval) v = 0;
945 : else
946 : {
947 300796 : long Nval = idealHNF_norm_pval(A, p, Zval);
948 300795 : v = idealHNF_val(A, P, Nval, Zval);
949 : }
950 834795 : return gc_long(av, vcA? v + vcA*pr_get_e(P): v);
951 : }
952 : GEN
953 7105 : gpidealval(GEN nf, GEN ix, GEN P)
954 : {
955 7105 : long v = idealval(nf,ix,P);
956 7105 : return v == LONG_MAX? mkoo(): stoi(v);
957 : }
958 :
959 : /* gcd and generalized Bezout */
960 :
961 : GEN
962 73630 : idealadd(GEN nf, GEN x, GEN y)
963 : {
964 73630 : pari_sp av = avma;
965 : long tx, ty;
966 : GEN z, a, dx, dy, dz;
967 :
968 73630 : tx = idealtyp(&x, NULL);
969 73630 : ty = idealtyp(&y, NULL); nf = checknf(nf);
970 73630 : if (tx != id_MAT) x = idealhnf_shallow(nf,x);
971 73630 : if (ty != id_MAT) y = idealhnf_shallow(nf,y);
972 73630 : if (lg(x) == 1) return gerepilecopy(av,y);
973 73504 : if (lg(y) == 1) return gerepilecopy(av,x); /* check for 0 ideal */
974 73049 : dx = Q_denom(x);
975 73049 : dy = Q_denom(y); dz = lcmii(dx,dy);
976 73049 : if (is_pm1(dz)) dz = NULL; else {
977 13083 : x = Q_muli_to_int(x, dz);
978 13083 : y = Q_muli_to_int(y, dz);
979 : }
980 73049 : a = gcdii(gcoeff(x,1,1), gcoeff(y,1,1));
981 73049 : if (is_pm1(a))
982 : {
983 34601 : long N = lg(x)-1;
984 34601 : if (!dz) { set_avma(av); return matid(N); }
985 3794 : return gerepileupto(av, scalarmat(ginv(dz), N));
986 : }
987 38448 : z = ZM_hnfmodid(shallowconcat(x,y), a);
988 38448 : if (dz) z = RgM_Rg_div(z,dz);
989 38448 : return gerepileupto(av,z);
990 : }
991 :
992 : static GEN
993 28 : trivial_merge(GEN x)
994 28 : { return (lg(x) == 1 || !is_pm1(gcoeff(x,1,1)))? NULL: gen_1; }
995 : /* true nf */
996 : static GEN
997 621833 : _idealaddtoone(GEN nf, GEN x, GEN y, long red)
998 : {
999 : GEN a;
1000 621833 : long tx = idealtyp(&x, NULL);
1001 621819 : long ty = idealtyp(&y, NULL);
1002 : long ea;
1003 621823 : if (tx != id_MAT) x = idealhnf_shallow(nf, x);
1004 621852 : if (ty != id_MAT) y = idealhnf_shallow(nf, y);
1005 621852 : if (lg(x) == 1)
1006 14 : a = trivial_merge(y);
1007 621838 : else if (lg(y) == 1)
1008 14 : a = trivial_merge(x);
1009 : else
1010 621824 : a = hnfmerge_get_1(x, y);
1011 621787 : if (!a) pari_err_COPRIME("idealaddtoone",x,y);
1012 621768 : if (red && (ea = gexpo(a)) > 10)
1013 : {
1014 5425 : GEN b = (typ(a) == t_COL)? a: scalarcol_shallow(a, nf_get_degree(nf));
1015 5425 : b = ZC_reducemodlll(b, idealHNF_mul(nf,x,y));
1016 5425 : if (gexpo(b) < ea) a = b;
1017 : }
1018 621768 : return a;
1019 : }
1020 : /* true nf */
1021 : GEN
1022 18137 : idealaddtoone_i(GEN nf, GEN x, GEN y)
1023 18137 : { return _idealaddtoone(nf, x, y, 1); }
1024 : /* true nf */
1025 : GEN
1026 603698 : idealaddtoone_raw(GEN nf, GEN x, GEN y)
1027 603698 : { return _idealaddtoone(nf, x, y, 0); }
1028 :
1029 : GEN
1030 98 : idealaddtoone(GEN nf, GEN x, GEN y)
1031 : {
1032 98 : GEN z = cgetg(3,t_VEC), a;
1033 98 : pari_sp av = avma;
1034 98 : nf = checknf(nf);
1035 98 : a = gerepileupto(av, idealaddtoone_i(nf,x,y));
1036 84 : gel(z,1) = a;
1037 84 : gel(z,2) = typ(a) == t_COL? Z_ZC_sub(gen_1,a): subui(1,a);
1038 84 : return z;
1039 : }
1040 :
1041 : /* assume elements of list are integral ideals */
1042 : GEN
1043 35 : idealaddmultoone(GEN nf, GEN list)
1044 : {
1045 35 : pari_sp av = avma;
1046 35 : long N, i, l, nz, tx = typ(list);
1047 : GEN H, U, perm, L;
1048 :
1049 35 : nf = checknf(nf); N = nf_get_degree(nf);
1050 35 : if (!is_vec_t(tx)) pari_err_TYPE("idealaddmultoone",list);
1051 35 : l = lg(list);
1052 35 : L = cgetg(l, t_VEC);
1053 35 : if (l == 1)
1054 0 : pari_err_DOMAIN("idealaddmultoone", "sum(ideals)", "!=", gen_1, L);
1055 35 : nz = 0; /* number of nonzero ideals in L */
1056 98 : for (i=1; i<l; i++)
1057 : {
1058 70 : GEN I = gel(list,i);
1059 70 : if (typ(I) != t_MAT) I = idealhnf_shallow(nf,I);
1060 70 : if (lg(I) != 1)
1061 : {
1062 42 : nz++; RgM_check_ZM(I,"idealaddmultoone");
1063 35 : if (lgcols(I) != N+1) pari_err_TYPE("idealaddmultoone [not an ideal]", I);
1064 : }
1065 63 : gel(L,i) = I;
1066 : }
1067 28 : H = ZM_hnfperm(shallowconcat1(L), &U, &perm);
1068 28 : if (lg(H) == 1 || !equali1(gcoeff(H,1,1)))
1069 7 : pari_err_DOMAIN("idealaddmultoone", "sum(ideals)", "!=", gen_1, L);
1070 49 : for (i=1; i<=N; i++)
1071 49 : if (perm[i] == 1) break;
1072 21 : U = gel(U,(nz-1)*N + i); /* (L[1]|...|L[nz]) U = 1 */
1073 21 : nz = 0;
1074 63 : for (i=1; i<l; i++)
1075 : {
1076 42 : GEN c = gel(L,i);
1077 42 : if (lg(c) == 1)
1078 14 : c = gen_0;
1079 : else {
1080 28 : c = ZM_ZC_mul(c, vecslice(U, nz*N + 1, (nz+1)*N));
1081 28 : nz++;
1082 : }
1083 42 : gel(L,i) = c;
1084 : }
1085 21 : return gerepilecopy(av, L);
1086 : }
1087 :
1088 : /* multiplication */
1089 :
1090 : /* x integral ideal (without archimedean component) in HNF form
1091 : * y = [a,alpha] corresponds to the integral ideal aZ_K+alpha Z_K, a in Z,
1092 : * alpha a ZV or a ZM (multiplication table). Multiply them */
1093 : static GEN
1094 1036511 : idealHNF_mul_two(GEN nf, GEN x, GEN y)
1095 : {
1096 1036511 : GEN m, a = gel(y,1), alpha = gel(y,2);
1097 : long i, N;
1098 :
1099 1036511 : if (typ(alpha) != t_MAT)
1100 : {
1101 702983 : alpha = zk_scalar_or_multable(nf, alpha);
1102 702987 : if (typ(alpha) == t_INT) /* e.g. y inert ? 0 should not (but may) occur */
1103 18073 : return signe(a)? ZM_Z_mul(x, gcdii(a, alpha)): cgetg(1,t_MAT);
1104 : }
1105 1018442 : N = lg(x)-1; m = cgetg((N<<1)+1,t_MAT);
1106 4939776 : for (i=1; i<=N; i++) gel(m,i) = ZM_ZC_mul(alpha,gel(x,i));
1107 4938592 : for (i=1; i<=N; i++) gel(m,i+N) = ZC_Z_mul(gel(x,i), a);
1108 1017471 : return ZM_hnfmodid(m, mulii(a, gcoeff(x,1,1)));
1109 : }
1110 :
1111 : /* Assume x and y are integral in HNF form [NOT extended]. Not memory clean.
1112 : * HACK: ideal in y can be of the form [a,b], a in Z, b in Z_K */
1113 : GEN
1114 541841 : idealHNF_mul(GEN nf, GEN x, GEN y)
1115 : {
1116 : GEN z;
1117 541841 : if (typ(y) == t_VEC)
1118 211855 : z = idealHNF_mul_two(nf,x,y);
1119 : else
1120 : { /* reduce one ideal to two-elt form. The smallest */
1121 329986 : GEN xZ = gcoeff(x,1,1), yZ = gcoeff(y,1,1);
1122 329986 : if (cmpii(xZ, yZ) < 0)
1123 : {
1124 116815 : if (is_pm1(xZ)) return gcopy(y);
1125 99616 : z = idealHNF_mul_two(nf, y, mat_ideal_two_elt(nf,x));
1126 : }
1127 : else
1128 : {
1129 213172 : if (is_pm1(yZ)) return gcopy(x);
1130 107832 : z = idealHNF_mul_two(nf, x, mat_ideal_two_elt(nf,y));
1131 : }
1132 : }
1133 419304 : return z;
1134 : }
1135 :
1136 : /* operations on elements in factored form */
1137 :
1138 : GEN
1139 233193 : famat_mul_shallow(GEN f, GEN g)
1140 : {
1141 233193 : if (typ(f) != t_MAT) f = to_famat_shallow(f,gen_1);
1142 233193 : if (typ(g) != t_MAT) g = to_famat_shallow(g,gen_1);
1143 233193 : if (lgcols(f) == 1) return g;
1144 180950 : if (lgcols(g) == 1) return f;
1145 179233 : return mkmat2(shallowconcat(gel(f,1), gel(g,1)),
1146 179228 : shallowconcat(gel(f,2), gel(g,2)));
1147 : }
1148 : GEN
1149 88979 : famat_mulpow_shallow(GEN f, GEN g, GEN e)
1150 : {
1151 88979 : if (!signe(e)) return f;
1152 58063 : return famat_mul_shallow(f, famat_pow_shallow(g, e));
1153 : }
1154 :
1155 : GEN
1156 146269 : famat_mulpows_shallow(GEN f, GEN g, long e)
1157 : {
1158 146269 : if (e==0) return f;
1159 123007 : return famat_mul_shallow(f, famat_pows_shallow(g, e));
1160 : }
1161 :
1162 : GEN
1163 10143 : famat_div_shallow(GEN f, GEN g)
1164 10143 : { return famat_mul_shallow(f, famat_inv_shallow(g)); }
1165 :
1166 : GEN
1167 0 : to_famat(GEN x, GEN y) { retmkmat2(mkcolcopy(x), mkcolcopy(y)); }
1168 : GEN
1169 1492605 : to_famat_shallow(GEN x, GEN y) { return mkmat2(mkcol(x), mkcol(y)); }
1170 :
1171 : /* concat the single elt x; not gconcat since x may be a t_COL */
1172 : static GEN
1173 185786 : append(GEN v, GEN x)
1174 : {
1175 185786 : long i, l = lg(v);
1176 185786 : GEN w = cgetg(l+1, typ(v));
1177 1134825 : for (i=1; i<l; i++) gel(w,i) = gcopy(gel(v,i));
1178 185786 : gel(w,i) = gcopy(x); return w;
1179 : }
1180 : /* add x^1 to famat f */
1181 : static GEN
1182 171498 : famat_add(GEN f, GEN x)
1183 : {
1184 171498 : GEN h = cgetg(3,t_MAT);
1185 171498 : if (lgcols(f) == 1)
1186 : {
1187 10116 : gel(h,1) = mkcolcopy(x);
1188 10116 : gel(h,2) = mkcol(gen_1);
1189 : }
1190 : else
1191 : {
1192 161382 : gel(h,1) = append(gel(f,1), x);
1193 161382 : gel(h,2) = gconcat(gel(f,2), gen_1);
1194 : }
1195 171498 : return h;
1196 : }
1197 : /* add x^-1 to famat f */
1198 : static GEN
1199 41520 : famat_sub(GEN f, GEN x)
1200 : {
1201 41520 : GEN h = cgetg(3,t_MAT);
1202 41520 : if (lgcols(f) == 1)
1203 : {
1204 17116 : gel(h,1) = mkcolcopy(x);
1205 17116 : gel(h,2) = mkcol(gen_m1);
1206 : }
1207 : else
1208 : {
1209 24404 : gel(h,1) = append(gel(f,1), x);
1210 24404 : gel(h,2) = gconcat(gel(f,2), gen_m1);
1211 : }
1212 41520 : return h;
1213 : }
1214 :
1215 : GEN
1216 266568 : famat_mul(GEN f, GEN g)
1217 : {
1218 : GEN h;
1219 266568 : if (typ(g) != t_MAT) {
1220 43713 : if (typ(f) == t_MAT) return famat_add(f, g);
1221 0 : h = cgetg(3, t_MAT);
1222 0 : gel(h,1) = mkcol2(gcopy(f), gcopy(g));
1223 0 : gel(h,2) = mkcol2(gen_1, gen_1);
1224 : }
1225 222855 : if (typ(f) != t_MAT) return famat_add(g, f);
1226 95070 : if (lgcols(f) == 1) return gcopy(g);
1227 72532 : if (lgcols(g) == 1) return gcopy(f);
1228 68419 : h = cgetg(3,t_MAT);
1229 68419 : gel(h,1) = gconcat(gel(f,1), gel(g,1));
1230 68419 : gel(h,2) = gconcat(gel(f,2), gel(g,2));
1231 68419 : return h;
1232 : }
1233 :
1234 : GEN
1235 41527 : famat_div(GEN f, GEN g)
1236 : {
1237 : GEN h;
1238 41527 : if (typ(g) != t_MAT) {
1239 41478 : if (typ(f) == t_MAT) return famat_sub(f, g);
1240 0 : h = cgetg(3, t_MAT);
1241 0 : gel(h,1) = mkcol2(gcopy(f), gcopy(g));
1242 0 : gel(h,2) = mkcol2(gen_1, gen_m1);
1243 : }
1244 49 : if (typ(f) != t_MAT) return famat_sub(g, f);
1245 7 : if (lgcols(f) == 1) return famat_inv(g);
1246 7 : if (lgcols(g) == 1) return gcopy(f);
1247 7 : h = cgetg(3,t_MAT);
1248 7 : gel(h,1) = gconcat(gel(f,1), gel(g,1));
1249 7 : gel(h,2) = gconcat(gel(f,2), gneg(gel(g,2)));
1250 7 : return h;
1251 : }
1252 :
1253 : GEN
1254 23137 : famat_sqr(GEN f)
1255 : {
1256 : GEN h;
1257 23137 : if (typ(f) != t_MAT) return to_famat(f,gen_2);
1258 23137 : if (lgcols(f) == 1) return gcopy(f);
1259 14307 : h = cgetg(3,t_MAT);
1260 14307 : gel(h,1) = gcopy(gel(f,1));
1261 14307 : gel(h,2) = gmul2n(gel(f,2),1);
1262 14307 : return h;
1263 : }
1264 :
1265 : GEN
1266 26697 : famat_inv_shallow(GEN f)
1267 : {
1268 26697 : if (typ(f) != t_MAT) return to_famat_shallow(f,gen_m1);
1269 10290 : if (lgcols(f) == 1) return f;
1270 10290 : return mkmat2(gel(f,1), ZC_neg(gel(f,2)));
1271 : }
1272 : GEN
1273 20358 : famat_inv(GEN f)
1274 : {
1275 20358 : if (typ(f) != t_MAT) return to_famat(f,gen_m1);
1276 20358 : if (lgcols(f) == 1) return gcopy(f);
1277 2150 : retmkmat2(gcopy(gel(f,1)), ZC_neg(gel(f,2)));
1278 : }
1279 : GEN
1280 60298 : famat_pow(GEN f, GEN n)
1281 : {
1282 60298 : if (typ(f) != t_MAT) return to_famat(f,n);
1283 60298 : if (lgcols(f) == 1) return gcopy(f);
1284 60298 : retmkmat2(gcopy(gel(f,1)), ZC_Z_mul(gel(f,2),n));
1285 : }
1286 : GEN
1287 67069 : famat_pow_shallow(GEN f, GEN n)
1288 : {
1289 67069 : if (is_pm1(n)) return signe(n) > 0? f: famat_inv_shallow(f);
1290 38421 : if (typ(f) != t_MAT) return to_famat_shallow(f,n);
1291 9948 : if (lgcols(f) == 1) return f;
1292 7393 : return mkmat2(gel(f,1), ZC_Z_mul(gel(f,2),n));
1293 : }
1294 :
1295 : GEN
1296 150327 : famat_pows_shallow(GEN f, long n)
1297 : {
1298 150327 : if (n==1) return f;
1299 63191 : if (n==-1) return famat_inv_shallow(f);
1300 63177 : if (typ(f) != t_MAT) return to_famat_shallow(f, stoi(n));
1301 26574 : if (lgcols(f) == 1) return f;
1302 26574 : return mkmat2(gel(f,1), ZC_z_mul(gel(f,2),n));
1303 : }
1304 :
1305 : GEN
1306 0 : famat_Z_gcd(GEN M, GEN n)
1307 : {
1308 0 : pari_sp av=avma;
1309 0 : long i, j, l=lgcols(M);
1310 0 : GEN F=cgetg(3,t_MAT);
1311 0 : gel(F,1)=cgetg(l,t_COL);
1312 0 : gel(F,2)=cgetg(l,t_COL);
1313 0 : for (i=1, j=1; i<l; i++)
1314 : {
1315 0 : GEN p = gcoeff(M,i,1);
1316 0 : GEN e = gminsg(Z_pval(n,p),gcoeff(M,i,2));
1317 0 : if (signe(e))
1318 : {
1319 0 : gcoeff(F,j,1)=p;
1320 0 : gcoeff(F,j,2)=e;
1321 0 : j++;
1322 : }
1323 : }
1324 0 : setlg(gel(F,1),j); setlg(gel(F,2),j);
1325 0 : return gerepilecopy(av,F);
1326 : }
1327 :
1328 : /* x assumed to be a t_MATs (factorization matrix), or compatible with
1329 : * the element_* functions. */
1330 : static GEN
1331 34071 : ext_sqr(GEN nf, GEN x)
1332 34071 : { return (typ(x)==t_MAT)? famat_sqr(x): nfsqr(nf, x); }
1333 : static GEN
1334 89822 : ext_mul(GEN nf, GEN x, GEN y)
1335 89822 : { return (typ(x)==t_MAT)? famat_mul(x,y): nfmul(nf, x, y); }
1336 : static GEN
1337 20358 : ext_inv(GEN nf, GEN x)
1338 20358 : { return (typ(x)==t_MAT)? famat_inv(x): nfinv(nf, x); }
1339 : static GEN
1340 0 : ext_pow(GEN nf, GEN x, GEN n)
1341 0 : { return (typ(x)==t_MAT)? famat_pow(x,n): nfpow(nf, x, n); }
1342 :
1343 : GEN
1344 0 : famat_to_nf(GEN nf, GEN f)
1345 : {
1346 : GEN t, x, e;
1347 : long i;
1348 0 : if (lgcols(f) == 1) return gen_1;
1349 0 : x = gel(f,1);
1350 0 : e = gel(f,2);
1351 0 : t = nfpow(nf, gel(x,1), gel(e,1));
1352 0 : for (i=lg(x)-1; i>1; i--)
1353 0 : t = nfmul(nf, t, nfpow(nf, gel(x,i), gel(e,i)));
1354 0 : return t;
1355 : }
1356 :
1357 : GEN
1358 0 : famat_idealfactor(GEN nf, GEN x)
1359 : {
1360 : long i, l;
1361 0 : GEN g = gel(x,1), e = gel(x,2), h = cgetg_copy(g, &l);
1362 0 : for (i = 1; i < l; i++) gel(h,i) = idealfactor(nf, gel(g,i));
1363 0 : h = famat_reduce(famatV_factorback(h,e));
1364 0 : return sort_factor(h, (void*)&cmp_prime_ideal, &cmp_nodata);
1365 : }
1366 :
1367 : GEN
1368 324165 : famat_reduce(GEN fa)
1369 : {
1370 : GEN E, G, L, g, e;
1371 : long i, k, l;
1372 :
1373 324165 : if (typ(fa) != t_MAT || lgcols(fa) == 1) return fa;
1374 311893 : g = gel(fa,1); l = lg(g);
1375 311893 : e = gel(fa,2);
1376 311893 : L = gen_indexsort(g, (void*)&cmp_universal, &cmp_nodata);
1377 311897 : G = cgetg(l, t_COL);
1378 311895 : E = cgetg(l, t_COL);
1379 : /* merge */
1380 2652243 : for (k=i=1; i<l; i++,k++)
1381 : {
1382 2340347 : gel(G,k) = gel(g,L[i]);
1383 2340347 : gel(E,k) = gel(e,L[i]);
1384 2340347 : if (k > 1 && gidentical(gel(G,k), gel(G,k-1)))
1385 : {
1386 1075934 : gel(E,k-1) = addii(gel(E,k), gel(E,k-1));
1387 1075931 : k--;
1388 : }
1389 : }
1390 : /* kill 0 exponents */
1391 311896 : l = k;
1392 1576316 : for (k=i=1; i<l; i++)
1393 1264419 : if (!gequal0(gel(E,i)))
1394 : {
1395 1252414 : gel(G,k) = gel(G,i);
1396 1252414 : gel(E,k) = gel(E,i); k++;
1397 : }
1398 311897 : setlg(G, k);
1399 311896 : setlg(E, k); return mkmat2(G,E);
1400 : }
1401 : GEN
1402 63 : matreduce(GEN f)
1403 63 : { pari_sp av = avma;
1404 63 : switch(typ(f))
1405 : {
1406 21 : case t_VEC: case t_COL:
1407 : {
1408 21 : GEN e; f = vec_reduce(f, &e); settyp(f, t_COL);
1409 21 : return gerepilecopy(av, mkmat2(f, zc_to_ZC(e)));
1410 : }
1411 35 : case t_MAT:
1412 35 : if (lg(f) == 3) break;
1413 : default:
1414 14 : pari_err_TYPE("matreduce", f);
1415 : }
1416 28 : if (typ(gel(f,1)) == t_VECSMALL)
1417 0 : f = famatsmall_reduce(f);
1418 : else
1419 : {
1420 28 : if (!RgV_is_ZV(gel(f,2))) pari_err_TYPE("matreduce",f);
1421 21 : f = famat_reduce(f);
1422 : }
1423 21 : return gerepilecopy(av, f);
1424 : }
1425 :
1426 : GEN
1427 14679 : famatsmall_reduce(GEN fa)
1428 : {
1429 : GEN E, G, L, g, e;
1430 : long i, k, l;
1431 14679 : if (lgcols(fa) == 1) return fa;
1432 14679 : g = gel(fa,1); l = lg(g);
1433 14679 : e = gel(fa,2);
1434 14679 : L = vecsmall_indexsort(g);
1435 14679 : G = cgetg(l, t_VECSMALL);
1436 14679 : E = cgetg(l, t_VECSMALL);
1437 : /* merge */
1438 131238 : for (k=i=1; i<l; i++,k++)
1439 : {
1440 116559 : G[k] = g[L[i]];
1441 116559 : E[k] = e[L[i]];
1442 116559 : if (k > 1 && G[k] == G[k-1])
1443 : {
1444 7075 : E[k-1] += E[k];
1445 7075 : k--;
1446 : }
1447 : }
1448 : /* kill 0 exponents */
1449 14679 : l = k;
1450 124163 : for (k=i=1; i<l; i++)
1451 109484 : if (E[i])
1452 : {
1453 105733 : G[k] = G[i];
1454 105733 : E[k] = E[i]; k++;
1455 : }
1456 14679 : setlg(G, k);
1457 14679 : setlg(E, k); return mkmat2(G,E);
1458 : }
1459 :
1460 : GEN
1461 55057 : famat_remove_trivial(GEN fa)
1462 : {
1463 55057 : GEN P, E, p = gel(fa,1), e = gel(fa,2);
1464 55057 : long j, k, l = lg(p);
1465 55057 : P = cgetg(l, t_COL);
1466 55057 : E = cgetg(l, t_COL);
1467 1801725 : for (j = k = 1; j < l; j++)
1468 1746668 : if (signe(gel(e,j))) { gel(P,k) = gel(p,j); gel(E,k++) = gel(e,j); }
1469 55057 : setlg(P, k); setlg(E, k); return mkmat2(P,E);
1470 : }
1471 :
1472 : GEN
1473 15605 : famatV_factorback(GEN v, GEN e)
1474 : {
1475 15605 : long i, l = lg(e);
1476 : GEN V;
1477 15605 : if (l == 1) return trivial_fact();
1478 15220 : V = signe(gel(e,1))? famat_pow_shallow(gel(v,1), gel(e,1)): trivial_fact();
1479 58500 : for (i = 2; i < l; i++) V = famat_mulpow_shallow(V, gel(v,i), gel(e,i));
1480 15220 : return V;
1481 : }
1482 :
1483 : GEN
1484 46529 : famatV_zv_factorback(GEN v, GEN e)
1485 : {
1486 46529 : long i, l = lg(e);
1487 : GEN V;
1488 46529 : if (l == 1) return trivial_fact();
1489 44142 : V = uel(e,1)? famat_pows_shallow(gel(v,1), uel(e,1)): trivial_fact();
1490 138698 : for (i = 2; i < l; i++) V = famat_mulpows_shallow(V, gel(v,i), uel(e,i));
1491 44142 : return V;
1492 : }
1493 :
1494 : GEN
1495 159821 : ZM_famat_limit(GEN fa, GEN limit)
1496 : {
1497 : pari_sp av;
1498 : GEN E, G, g, e, r;
1499 : long i, k, l, n, lG;
1500 :
1501 159821 : if (lgcols(fa) == 1) return fa;
1502 159814 : g = gel(fa,1); l = lg(g);
1503 159814 : e = gel(fa,2);
1504 320778 : for(n=0, i=1; i<l; i++)
1505 160963 : if (cmpii(gel(g,i),limit)<=0) n++;
1506 159815 : lG = n<l-1 ? n+2 : n+1;
1507 159815 : G = cgetg(lG, t_COL);
1508 159816 : E = cgetg(lG, t_COL);
1509 159816 : av = avma;
1510 320780 : for (i=1, k=1, r = gen_1; i<l; i++)
1511 : {
1512 160964 : if (cmpii(gel(g,i),limit)<=0)
1513 : {
1514 160831 : gel(G,k) = gel(g,i);
1515 160831 : gel(E,k) = gel(e,i);
1516 160831 : k++;
1517 133 : } else r = mulii(r, powii(gel(g,i), gel(e,i)));
1518 : }
1519 159816 : if (k<i)
1520 : {
1521 133 : gel(G, k) = gerepileuptoint(av, r);
1522 133 : gel(E, k) = gen_1;
1523 : }
1524 159816 : return mkmat2(G,E);
1525 : }
1526 :
1527 : /* assume pr has degree 1 and coprime to Q_denom(x) */
1528 : static GEN
1529 119299 : to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1530 : {
1531 119299 : GEN d, r, p = modpr_get_p(modpr);
1532 119299 : x = nf_to_scalar_or_basis(nf,x);
1533 119299 : if (typ(x) != t_COL) return Rg_to_Fp(x,p);
1534 118151 : x = Q_remove_denom(x, &d);
1535 118151 : r = zk_to_Fq(x, modpr);
1536 118148 : if (d) r = Fp_div(r, d, p);
1537 118148 : return r;
1538 : }
1539 :
1540 : /* pr coprime to all denominators occurring in x */
1541 : static GEN
1542 602 : famat_to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1543 : {
1544 602 : GEN p = modpr_get_p(modpr);
1545 602 : GEN t = NULL, g = gel(x,1), e = gel(x,2), q = subiu(p,1);
1546 602 : long i, l = lg(g);
1547 3094 : for (i = 1; i < l; i++)
1548 : {
1549 2492 : GEN n = modii(gel(e,i), q);
1550 2492 : if (signe(n))
1551 : {
1552 2492 : GEN h = to_Fp_coprime(nf, gel(g,i), modpr);
1553 2492 : h = Fp_pow(h, n, p);
1554 2492 : t = t? Fp_mul(t, h, p): h;
1555 : }
1556 : }
1557 602 : return t? modii(t, p): gen_1;
1558 : }
1559 :
1560 : /* cf famat_to_nf_modideal_coprime, modpr attached to prime of degree 1 */
1561 : GEN
1562 117409 : nf_to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1563 : {
1564 602 : return typ(x)==t_MAT? famat_to_Fp_coprime(nf, x, modpr)
1565 118011 : : to_Fp_coprime(nf, x, modpr);
1566 : }
1567 :
1568 : static long
1569 2810391 : zk_pvalrem(GEN x, GEN p, GEN *py)
1570 2810391 : { return (typ(x) == t_INT)? Z_pvalrem(x, p, py): ZV_pvalrem(x, p, py); }
1571 : /* x a QC or Q. Return a ZC or Z, whose content is coprime to Z. Set v, dx
1572 : * such that x = p^v (newx / dx); dx = NULL if 1 */
1573 : static GEN
1574 2907427 : nf_remove_denom_p(GEN nf, GEN x, GEN p, GEN *pdx, long *pv)
1575 : {
1576 : long vcx;
1577 : GEN dx;
1578 2907427 : x = nf_to_scalar_or_basis(nf, x);
1579 2907426 : x = Q_remove_denom(x, &dx);
1580 2907433 : if (dx)
1581 : {
1582 351738 : vcx = - Z_pvalrem(dx, p, &dx);
1583 351738 : if (!vcx) vcx = zk_pvalrem(x, p, &x);
1584 351738 : if (isint1(dx)) dx = NULL;
1585 : }
1586 : else
1587 : {
1588 2555695 : vcx = zk_pvalrem(x, p, &x);
1589 2555698 : dx = NULL;
1590 : }
1591 2907436 : *pv = vcx;
1592 2907436 : *pdx = dx; return x;
1593 : }
1594 : /* x = b^e/p^(e-1) in Z_K; x = 0 mod p/pr^e, (x,pr) = 1. Return NULL
1595 : * if p inert (instead of 1) */
1596 : static GEN
1597 89519 : p_makecoprime(GEN pr)
1598 : {
1599 89519 : GEN B = pr_get_tau(pr), b;
1600 : long i, e;
1601 :
1602 89519 : if (typ(B) == t_INT) return NULL;
1603 66902 : b = gel(B,1); /* B = multiplication table by b */
1604 66902 : e = pr_get_e(pr);
1605 66902 : if (e == 1) return b;
1606 : /* one could also divide (exactly) by p in each iteration */
1607 40331 : for (i = 1; i < e; i++) b = ZM_ZC_mul(B, b);
1608 19754 : return ZC_Z_divexact(b, powiu(pr_get_p(pr), e-1));
1609 : }
1610 :
1611 : /* Compute A = prod g[i]^e[i] mod pr^k, assuming (A, pr) = 1.
1612 : * Method: modify each g[i] so that it becomes coprime to pr,
1613 : * g[i] *= (b/p)^v_pr(g[i]), where b/p = pr^(-1) times something integral
1614 : * and prime to p; globally, we multiply by (b/p)^v_pr(A) = 1.
1615 : * Optimizations:
1616 : * 1) remove all powers of p from contents, and consider extra generator p^vp;
1617 : * modified as p * (b/p)^e = b^e / p^(e-1)
1618 : * 2) remove denominators, coprime to p, by multiplying by inverse mod prk\cap Z
1619 : *
1620 : * EX = multiple of exponent of (O_K / pr^k)^* used to reduce the product in
1621 : * case the e[i] are large */
1622 : GEN
1623 1000181 : famat_makecoprime(GEN nf, GEN g, GEN e, GEN pr, GEN prk, GEN EX)
1624 : {
1625 1000181 : GEN G, E, t, vp = NULL, p = pr_get_p(pr), prkZ = gcoeff(prk, 1,1);
1626 1000181 : long i, l = lg(g);
1627 :
1628 1000181 : G = cgetg(l+1, t_VEC);
1629 1000192 : E = cgetg(l+1, t_VEC); /* l+1: room for "modified p" */
1630 3907603 : for (i=1; i < l; i++)
1631 : {
1632 : long vcx;
1633 2907424 : GEN dx, x = nf_remove_denom_p(nf, gel(g,i), p, &dx, &vcx);
1634 2907435 : if (vcx) /* = v_p(content(g[i])) */
1635 : {
1636 135485 : GEN a = mulsi(vcx, gel(e,i));
1637 135486 : vp = vp? addii(vp, a): a;
1638 : }
1639 : /* x integral, content coprime to p; dx coprime to p */
1640 2907436 : if (typ(x) == t_INT)
1641 : { /* x coprime to p, hence to pr */
1642 616861 : x = modii(x, prkZ);
1643 616860 : if (dx) x = Fp_div(x, dx, prkZ);
1644 : }
1645 : else
1646 : {
1647 2290575 : (void)ZC_nfvalrem(x, pr, &x); /* x *= (b/p)^v_pr(x) */
1648 2290530 : x = ZC_hnfrem(FpC_red(x,prkZ), prk);
1649 2290557 : if (dx) x = FpC_Fp_mul(x, Fp_inv(dx,prkZ), prkZ);
1650 : }
1651 2907410 : gel(G,i) = x;
1652 2907410 : gel(E,i) = gel(e,i);
1653 : }
1654 :
1655 1000179 : t = vp? p_makecoprime(pr): NULL;
1656 1000183 : if (!t)
1657 : { /* no need for extra generator */
1658 933302 : setlg(G,l);
1659 933307 : setlg(E,l);
1660 : }
1661 : else
1662 : {
1663 66881 : gel(G,i) = FpC_red(t, prkZ);
1664 66881 : gel(E,i) = vp;
1665 : }
1666 1000189 : return famat_to_nf_modideal_coprime(nf, G, E, prk, EX);
1667 : }
1668 :
1669 : /* simplified version of famat_makecoprime for X = SUnits[1] */
1670 : GEN
1671 91 : sunits_makecoprime(GEN X, GEN pr, GEN prk)
1672 : {
1673 91 : GEN G, p = pr_get_p(pr), prkZ = gcoeff(prk,1,1);
1674 91 : long i, l = lg(X);
1675 :
1676 91 : G = cgetg(l, t_VEC);
1677 5992 : for (i = 1; i < l; i++)
1678 : {
1679 5901 : GEN x = gel(X,i);
1680 5901 : if (typ(x) == t_INT) /* a prime */
1681 1281 : x = equalii(x,p)? p_makecoprime(pr): modii(x, prkZ);
1682 : else
1683 : {
1684 4620 : (void)ZC_nfvalrem(x, pr, &x); /* x *= (b/p)^v_pr(x) */
1685 4620 : x = ZC_hnfrem(FpC_red(x,prkZ), prk);
1686 : }
1687 5901 : gel(G,i) = x;
1688 : }
1689 91 : return G;
1690 : }
1691 :
1692 : /* prod g[i]^e[i] mod bid, assume (g[i], id) = 1 and 1 < lg(g) <= lg(e) */
1693 : GEN
1694 19845 : famat_to_nf_moddivisor(GEN nf, GEN g, GEN e, GEN bid)
1695 : {
1696 19845 : GEN t, cyc = bid_get_cyc(bid);
1697 19845 : if (lg(cyc) == 1)
1698 0 : t = gen_1;
1699 : else
1700 19845 : t = famat_to_nf_modideal_coprime(nf, g, e, bid_get_ideal(bid),
1701 : cyc_get_expo(cyc));
1702 19845 : return set_sign_mod_divisor(nf, mkmat2(g,e), t, bid_get_sarch(bid));
1703 : }
1704 :
1705 : GEN
1706 765697 : vecmul(GEN x, GEN y)
1707 : {
1708 765697 : if (is_scalar_t(typ(x))) return gmul(x,y);
1709 765694 : pari_APPLY_same(vecmul(gel(x,i), gel(y,i)))
1710 : }
1711 :
1712 : GEN
1713 0 : vecinv(GEN x)
1714 : {
1715 0 : if (is_scalar_t(typ(x))) return ginv(x);
1716 0 : pari_APPLY_same(vecinv(gel(x,i)))
1717 : }
1718 :
1719 : GEN
1720 0 : vecpow(GEN x, GEN n)
1721 : {
1722 0 : if (is_scalar_t(typ(x))) return powgi(x,n);
1723 0 : pari_APPLY_same(vecpow(gel(x,i), n))
1724 : }
1725 :
1726 : GEN
1727 903 : vecdiv(GEN x, GEN y)
1728 : {
1729 903 : if (is_scalar_t(typ(x))) return gdiv(x,y);
1730 903 : pari_APPLY_same(vecdiv(gel(x,i), gel(y,i)))
1731 : }
1732 :
1733 : /* A ideal as a square t_MAT */
1734 : static GEN
1735 295269 : idealmulelt(GEN nf, GEN x, GEN A)
1736 : {
1737 : long i, lx;
1738 : GEN dx, dA, D;
1739 295269 : if (lg(A) == 1) return cgetg(1, t_MAT);
1740 295269 : x = nf_to_scalar_or_basis(nf,x);
1741 295269 : if (typ(x) != t_COL)
1742 92094 : return isintzero(x)? cgetg(1,t_MAT): RgM_Rg_mul(A, Q_abs_shallow(x));
1743 203175 : x = Q_remove_denom(x, &dx);
1744 203175 : A = Q_remove_denom(A, &dA);
1745 203175 : x = zk_multable(nf, x);
1746 203175 : D = mulii(zkmultable_capZ(x), gcoeff(A,1,1));
1747 203175 : x = zkC_multable_mul(A, x);
1748 203175 : settyp(x, t_MAT); lx = lg(x);
1749 : /* x may contain scalars (at most 1 since the ideal is nonzero)*/
1750 773487 : for (i=1; i<lx; i++)
1751 582597 : if (typ(gel(x,i)) == t_INT)
1752 : {
1753 12285 : if (i > 1) swap(gel(x,1), gel(x,i)); /* help HNF */
1754 12285 : gel(x,1) = scalarcol_shallow(gel(x,1), lx-1);
1755 12285 : break;
1756 : }
1757 203175 : x = ZM_hnfmodid(x, D);
1758 203175 : dx = mul_denom(dx,dA);
1759 203175 : return dx? gdiv(x,dx): x;
1760 : }
1761 :
1762 : /* nf a true nf, tx <= ty */
1763 : static GEN
1764 936952 : idealmul_aux(GEN nf, GEN x, GEN y, long tx, long ty)
1765 : {
1766 : GEN z, cx, cy;
1767 936952 : switch(tx)
1768 : {
1769 348952 : case id_PRINCIPAL:
1770 348952 : switch(ty)
1771 : {
1772 53291 : case id_PRINCIPAL:
1773 53291 : return idealhnf_principal(nf, nfmul(nf,x,y));
1774 392 : case id_PRIME:
1775 : {
1776 392 : GEN p = pr_get_p(y), pi = pr_get_gen(y), cx;
1777 392 : if (pr_is_inert(y)) return RgM_Rg_mul(idealhnf_principal(nf,x),p);
1778 :
1779 217 : x = nf_to_scalar_or_basis(nf, x);
1780 217 : switch(typ(x))
1781 : {
1782 203 : case t_INT:
1783 203 : if (!signe(x)) return cgetg(1,t_MAT);
1784 203 : return ZM_Z_mul(pr_hnf(nf,y), absi_shallow(x));
1785 7 : case t_FRAC:
1786 7 : return RgM_Rg_mul(pr_hnf(nf,y), Q_abs_shallow(x));
1787 : }
1788 : /* t_COL */
1789 7 : x = Q_primitive_part(x, &cx);
1790 7 : x = zk_multable(nf, x);
1791 7 : z = shallowconcat(ZM_Z_mul(x,p), ZM_ZC_mul(x,pi));
1792 7 : z = ZM_hnfmodid(z, mulii(p, zkmultable_capZ(x)));
1793 7 : return cx? ZM_Q_mul(z, cx): z;
1794 : }
1795 295269 : default: /* id_MAT */
1796 295269 : return idealmulelt(nf, x,y);
1797 : }
1798 267296 : case id_PRIME:
1799 267296 : if (ty==id_PRIME)
1800 260901 : { y = pr_hnf(nf,y); cy = NULL; }
1801 : else
1802 6395 : y = Q_primitive_part(y, &cy);
1803 267296 : y = idealHNF_mul_two(nf,y,x);
1804 267296 : return cy? ZM_Q_mul(y,cy): y;
1805 :
1806 320704 : default: /* id_MAT */
1807 : {
1808 320704 : long N = nf_get_degree(nf);
1809 320704 : if (lg(x)-1 != N || lg(y)-1 != N) pari_err_DIM("idealmul");
1810 320690 : x = Q_primitive_part(x, &cx);
1811 320690 : y = Q_primitive_part(y, &cy); cx = mul_content(cx,cy);
1812 320690 : y = idealHNF_mul(nf,x,y);
1813 320690 : return cx? ZM_Q_mul(y,cx): y;
1814 : }
1815 : }
1816 : }
1817 :
1818 : /* output the ideal product x.y */
1819 : GEN
1820 936951 : idealmul(GEN nf, GEN x, GEN y)
1821 : {
1822 : pari_sp av;
1823 : GEN res, ax, ay, z;
1824 936951 : long tx = idealtyp(&x,&ax);
1825 936951 : long ty = idealtyp(&y,&ay), f;
1826 936951 : if (tx>ty) { swap(ax,ay); swap(x,y); lswap(tx,ty); }
1827 936951 : f = (ax||ay); res = f? cgetg(3,t_VEC): NULL; /*product is an extended ideal*/
1828 936951 : av = avma;
1829 936951 : z = gerepileupto(av, idealmul_aux(checknf(nf), x,y, tx,ty));
1830 936938 : if (!f) return z;
1831 27660 : if (ax && ay)
1832 26148 : ax = ext_mul(nf, ax, ay);
1833 : else
1834 1512 : ax = gcopy(ax? ax: ay);
1835 27660 : gel(res,1) = z; gel(res,2) = ax; return res;
1836 : }
1837 :
1838 : /* Return x, integral in 2-elt form, such that pr^2 = c * x. cf idealpowprime
1839 : * nf = true nf */
1840 : static GEN
1841 209069 : idealsqrprime(GEN nf, GEN pr, GEN *pc)
1842 : {
1843 209069 : GEN p = pr_get_p(pr), q, gen;
1844 209069 : long e = pr_get_e(pr), f = pr_get_f(pr);
1845 :
1846 209070 : q = (e == 1)? sqri(p): p;
1847 209063 : if (e <= 2 && e * f == nf_get_degree(nf))
1848 : { /* pr^e = (p) */
1849 44100 : *pc = q;
1850 44100 : return mkvec2(gen_1,gen_0);
1851 : }
1852 164969 : gen = nfsqr(nf, pr_get_gen(pr));
1853 164970 : gen = FpC_red(gen, q);
1854 164951 : *pc = NULL;
1855 164951 : return mkvec2(q, gen);
1856 : }
1857 : /* cf idealpow_aux */
1858 : static GEN
1859 38670 : idealsqr_aux(GEN nf, GEN x, long tx)
1860 : {
1861 38670 : GEN T = nf_get_pol(nf), m, cx, a, alpha;
1862 38670 : long N = degpol(T);
1863 38670 : switch(tx)
1864 : {
1865 70 : case id_PRINCIPAL:
1866 70 : return idealhnf_principal(nf, nfsqr(nf,x));
1867 10761 : case id_PRIME:
1868 10761 : if (pr_is_inert(x)) return scalarmat(sqri(gel(x,1)), N);
1869 10593 : x = idealsqrprime(nf, x, &cx);
1870 10593 : x = idealhnf_two(nf,x);
1871 10593 : return cx? ZM_Z_mul(x, cx): x;
1872 27839 : default:
1873 27839 : x = Q_primitive_part(x, &cx);
1874 27839 : a = mat_ideal_two_elt(nf,x); alpha = gel(a,2); a = gel(a,1);
1875 27839 : alpha = nfsqr(nf,alpha);
1876 27839 : m = zk_scalar_or_multable(nf, alpha);
1877 27839 : if (typ(m) == t_INT) {
1878 1568 : x = gcdii(sqri(a), m);
1879 1568 : if (cx) x = gmul(x, gsqr(cx));
1880 1568 : x = scalarmat(x, N);
1881 : }
1882 : else
1883 : { /* could use gcdii(sqri(a), zkmultable_capZ(m)), but costly */
1884 26271 : x = ZM_hnfmodid(m, sqri(a));
1885 26271 : if (cx) cx = gsqr(cx);
1886 26271 : if (cx) x = ZM_Q_mul(x, cx);
1887 : }
1888 27839 : return x;
1889 : }
1890 : }
1891 : GEN
1892 38670 : idealsqr(GEN nf, GEN x)
1893 : {
1894 : pari_sp av;
1895 : GEN res, ax, z;
1896 38670 : long tx = idealtyp(&x,&ax);
1897 38670 : res = ax? cgetg(3,t_VEC): NULL; /*product is an extended ideal*/
1898 38670 : av = avma;
1899 38670 : z = gerepileupto(av, idealsqr_aux(checknf(nf), x, tx));
1900 38670 : if (!ax) return z;
1901 34071 : gel(res,1) = z;
1902 34071 : gel(res,2) = ext_sqr(nf, ax); return res;
1903 : }
1904 :
1905 : /* norm of an ideal */
1906 : GEN
1907 12825 : idealnorm(GEN nf, GEN x)
1908 : {
1909 : pari_sp av;
1910 : long tx;
1911 :
1912 12825 : switch(idealtyp(&x, NULL))
1913 : {
1914 630 : case id_PRIME: return pr_norm(x);
1915 9430 : case id_MAT: return RgM_det_triangular(x);
1916 : }
1917 : /* id_PRINCIPAL */
1918 2765 : nf = checknf(nf); av = avma;
1919 2765 : x = nfnorm(nf, x);
1920 2765 : tx = typ(x);
1921 2765 : if (tx == t_INT) return gerepileuptoint(av, absi(x));
1922 406 : if (tx != t_FRAC) pari_err_TYPE("idealnorm",x);
1923 406 : return gerepileupto(av, Q_abs(x));
1924 : }
1925 :
1926 : /* x \cap Z */
1927 : GEN
1928 2982 : idealdown(GEN nf, GEN x)
1929 : {
1930 2982 : pari_sp av = avma;
1931 : GEN y, c;
1932 2982 : switch(idealtyp(&x, NULL))
1933 : {
1934 7 : case id_PRIME: return icopy(pr_get_p(x));
1935 2107 : case id_MAT: return gcopy(gcoeff(x,1,1));
1936 : }
1937 : /* id_PRINCIPAL */
1938 868 : nf = checknf(nf); av = avma;
1939 868 : x = nf_to_scalar_or_basis(nf, x);
1940 868 : if (is_rational_t(typ(x))) return Q_abs(x);
1941 14 : x = Q_primitive_part(x, &c);
1942 14 : y = zkmultable_capZ(zk_multable(nf, x));
1943 14 : return gerepilecopy(av, mul_content(c, y));
1944 : }
1945 :
1946 : /* true nf */
1947 : static GEN
1948 35 : idealismaximal_int(GEN nf, GEN p)
1949 : {
1950 : GEN L;
1951 35 : if (!BPSW_psp(p)) return NULL;
1952 70 : if (!dvdii(nf_get_index(nf), p) &&
1953 49 : !FpX_is_irred(FpX_red(nf_get_pol(nf),p), p)) return NULL;
1954 21 : L = idealprimedec(nf, p);
1955 21 : return lg(L) == 2? gel(L,1): NULL;
1956 : }
1957 : /* true nf */
1958 : static GEN
1959 21 : idealismaximal_mat(GEN nf, GEN x)
1960 : {
1961 : GEN p, c, L;
1962 : long i, l, f;
1963 21 : x = Q_primitive_part(x, &c);
1964 21 : p = gcoeff(x,1,1);
1965 21 : if (c)
1966 : {
1967 7 : if (typ(c) == t_FRAC || !equali1(p)) return NULL;
1968 7 : return idealismaximal_int(nf, c);
1969 : }
1970 14 : if (!BPSW_psp(p)) return NULL;
1971 14 : l = lg(x); f = 1;
1972 35 : for (i = 2; i < l; i++)
1973 : {
1974 21 : c = gcoeff(x,i,i);
1975 21 : if (equalii(c, p)) f++; else if (!equali1(c)) return NULL;
1976 : }
1977 14 : L = idealprimedec_limit_f(nf, p, f);
1978 28 : for (i = lg(L)-1; i; i--)
1979 : {
1980 28 : GEN pr = gel(L,i);
1981 28 : if (pr_get_f(pr) != f) break;
1982 28 : if (idealval(nf, x, pr) == 1) return pr;
1983 : }
1984 0 : return NULL;
1985 : }
1986 : /* true nf */
1987 : static GEN
1988 56 : idealismaximal_i(GEN nf, GEN x)
1989 : {
1990 : GEN L, p, pr, c;
1991 : long i, l;
1992 56 : switch(idealtyp(&x, NULL))
1993 : {
1994 7 : case id_PRIME: return x;
1995 21 : case id_MAT: return idealismaximal_mat(nf, x);
1996 : }
1997 : /* id_PRINCIPAL */
1998 28 : x = nf_to_scalar_or_basis(nf, x);
1999 28 : switch(typ(x))
2000 : {
2001 28 : case t_INT: return idealismaximal_int(nf, absi_shallow(x));
2002 0 : case t_FRAC: return NULL;
2003 : }
2004 0 : x = Q_primitive_part(x, &c);
2005 0 : if (c) return NULL;
2006 0 : p = zkmultable_capZ(zk_multable(nf, x));
2007 0 : L = idealprimedec(nf, p); l = lg(L); pr = NULL;
2008 0 : for (i = 1; i < l; i++)
2009 : {
2010 0 : long v = ZC_nfval(x, gel(L,i));
2011 0 : if (v > 1 || (v && pr)) return NULL;
2012 0 : pr = gel(L,i);
2013 : }
2014 0 : return pr;
2015 : }
2016 : GEN
2017 56 : idealismaximal(GEN nf, GEN x)
2018 : {
2019 56 : pari_sp av = avma;
2020 56 : x = idealismaximal_i(checknf(nf), x);
2021 56 : if (!x) { set_avma(av); return gen_0; }
2022 42 : return gerepilecopy(av, x);
2023 : }
2024 :
2025 : /* I^(-1) = { x \in K, Tr(x D^(-1) I) \in Z }, D different of K/Q
2026 : *
2027 : * nf[5][6] = pp( D^(-1) ) = pp( HNF( T^(-1) ) ), T = (Tr(wi wj))
2028 : * nf[5][7] = same in 2-elt form.
2029 : * Assume I integral. Return the integral ideal (I\cap Z) I^(-1) */
2030 : GEN
2031 214792 : idealHNF_inv_Z(GEN nf, GEN I)
2032 : {
2033 214792 : GEN J, dual, IZ = gcoeff(I,1,1); /* I \cap Z */
2034 214792 : if (isint1(IZ)) return matid(lg(I)-1);
2035 195165 : J = idealHNF_mul(nf,I, gmael(nf,5,7));
2036 : /* I in HNF, hence easily inverted; multiply by IZ to get integer coeffs
2037 : * missing content cancels while solving the linear equation */
2038 195168 : dual = shallowtrans( hnf_divscale(J, gmael(nf,5,6), IZ) );
2039 195167 : return ZM_hnfmodid(dual, IZ);
2040 : }
2041 : /* I HNF with rational coefficients (denominator d). */
2042 : GEN
2043 73613 : idealHNF_inv(GEN nf, GEN I)
2044 : {
2045 73613 : GEN J, IQ = gcoeff(I,1,1); /* I \cap Q; d IQ = dI \cap Z */
2046 73613 : J = idealHNF_inv_Z(nf, Q_remove_denom(I, NULL)); /* = (dI)^(-1) * (d IQ) */
2047 73613 : return equali1(IQ)? J: RgM_Rg_div(J, IQ);
2048 : }
2049 :
2050 : /* return p * P^(-1) [integral] */
2051 : GEN
2052 36342 : pr_inv_p(GEN pr)
2053 : {
2054 36342 : if (pr_is_inert(pr)) return matid(pr_get_f(pr));
2055 35566 : return ZM_hnfmodid(pr_get_tau(pr), pr_get_p(pr));
2056 : }
2057 : GEN
2058 17823 : pr_inv(GEN pr)
2059 : {
2060 17823 : GEN p = pr_get_p(pr);
2061 17823 : if (pr_is_inert(pr)) return scalarmat(ginv(p), pr_get_f(pr));
2062 17431 : return RgM_Rg_div(ZM_hnfmodid(pr_get_tau(pr),p), p);
2063 : }
2064 :
2065 : GEN
2066 141626 : idealinv(GEN nf, GEN x)
2067 : {
2068 : GEN res, ax;
2069 : pari_sp av;
2070 141626 : long tx = idealtyp(&x,&ax), N;
2071 :
2072 141626 : res = ax? cgetg(3,t_VEC): NULL;
2073 141625 : nf = checknf(nf); av = avma;
2074 141625 : N = nf_get_degree(nf);
2075 141626 : switch (tx)
2076 : {
2077 68279 : case id_MAT:
2078 68279 : if (lg(x)-1 != N) pari_err_DIM("idealinv");
2079 68279 : x = idealHNF_inv(nf,x); break;
2080 56560 : case id_PRINCIPAL:
2081 56560 : x = nf_to_scalar_or_basis(nf, x);
2082 56560 : if (typ(x) != t_COL)
2083 56511 : x = idealhnf_principal(nf,ginv(x));
2084 : else
2085 : { /* nfinv + idealhnf where we already know (x) \cap Z */
2086 : GEN c, d;
2087 49 : x = Q_remove_denom(x, &c);
2088 49 : x = zk_inv(nf, x);
2089 49 : x = Q_remove_denom(x, &d); /* true inverse is c/d * x */
2090 49 : if (!d) /* x and x^(-1) integral => x a unit */
2091 14 : x = c? scalarmat(c, N): matid(N);
2092 : else
2093 : {
2094 35 : c = c? gdiv(c,d): ginv(d);
2095 35 : x = zk_multable(nf, x);
2096 35 : x = ZM_Q_mul(ZM_hnfmodid(x,d), c);
2097 : }
2098 : }
2099 56560 : break;
2100 16787 : case id_PRIME:
2101 16787 : x = pr_inv(x); break;
2102 : }
2103 141626 : x = gerepileupto(av,x); if (!ax) return x;
2104 20358 : gel(res,1) = x;
2105 20358 : gel(res,2) = ext_inv(nf, ax); return res;
2106 : }
2107 :
2108 : /* write x = A/B, A,B coprime integral ideals */
2109 : GEN
2110 285831 : idealnumden(GEN nf, GEN x)
2111 : {
2112 285831 : pari_sp av = avma;
2113 : GEN x0, c, d, A, B, J;
2114 285831 : long tx = idealtyp(&x, NULL);
2115 285830 : nf = checknf(nf);
2116 285835 : switch (tx)
2117 : {
2118 7 : case id_PRIME:
2119 7 : retmkvec2(idealhnf(nf, x), gen_1);
2120 51079 : case id_PRINCIPAL:
2121 : {
2122 : GEN xZ, mx;
2123 51079 : x = nf_to_scalar_or_basis(nf, x);
2124 51079 : switch(typ(x))
2125 : {
2126 2562 : case t_INT: return gerepilecopy(av, mkvec2(absi_shallow(x),gen_1));
2127 14 : case t_FRAC:return gerepilecopy(av, mkvec2(absi_shallow(gel(x,1)), gel(x,2)));
2128 : }
2129 : /* t_COL */
2130 48503 : x = Q_remove_denom(x, &d);
2131 48503 : if (!d) return gerepilecopy(av, mkvec2(idealhnf_shallow(nf, x), gen_1));
2132 91 : mx = zk_multable(nf, x);
2133 91 : xZ = zkmultable_capZ(mx);
2134 91 : x = ZM_hnfmodid(mx, xZ); /* principal ideal (x) */
2135 91 : x0 = mkvec2(xZ, mx); /* same, for fast multiplication */
2136 91 : break;
2137 : }
2138 234749 : default: /* id_MAT */
2139 : {
2140 234749 : long n = lg(x)-1;
2141 234749 : if (n == 0) return mkvec2(gen_0, gen_1);
2142 234749 : if (n != nf_get_degree(nf)) pari_err_DIM("idealnumden");
2143 234749 : x0 = x = Q_remove_denom(x, &d);
2144 234747 : if (!d) return gerepilecopy(av, mkvec2(x, gen_1));
2145 14 : break;
2146 : }
2147 : }
2148 105 : J = hnfmodid(x, d); /* = d/B */
2149 105 : c = gcoeff(J,1,1); /* (d/B) \cap Z, divides d */
2150 105 : B = idealHNF_inv_Z(nf, J); /* (d/B \cap Z) B/d */
2151 105 : if (!equalii(c,d)) B = ZM_Z_mul(B, diviiexact(d,c)); /* = B ! */
2152 105 : A = idealHNF_mul(nf, B, x0); /* d * (original x) * B = d A */
2153 105 : A = ZM_Z_divexact(A, d); /* = A ! */
2154 105 : return gerepilecopy(av, mkvec2(A, B));
2155 : }
2156 :
2157 : /* Return x, integral in 2-elt form, such that pr^n = c * x. Assume n != 0.
2158 : * nf = true nf */
2159 : static GEN
2160 930191 : idealpowprime(GEN nf, GEN pr, GEN n, GEN *pc)
2161 : {
2162 930191 : GEN p = pr_get_p(pr), q, gen;
2163 :
2164 930175 : *pc = NULL;
2165 930175 : if (is_pm1(n)) /* n = 1 special cased for efficiency */
2166 : {
2167 535664 : q = p;
2168 535664 : if (typ(pr_get_tau(pr)) == t_INT) /* inert */
2169 : {
2170 0 : *pc = (signe(n) >= 0)? p: ginv(p);
2171 0 : return mkvec2(gen_1,gen_0);
2172 : }
2173 535660 : if (signe(n) >= 0) gen = pr_get_gen(pr);
2174 : else
2175 : {
2176 155176 : gen = pr_get_tau(pr); /* possibly t_MAT */
2177 155185 : *pc = ginv(p);
2178 : }
2179 : }
2180 394569 : else if (equalis(n,2)) return idealsqrprime(nf, pr, pc);
2181 : else
2182 : {
2183 196093 : long e = pr_get_e(pr), f = pr_get_f(pr);
2184 196102 : GEN r, m = truedvmdis(n, e, &r);
2185 196100 : if (e * f == nf_get_degree(nf))
2186 : { /* pr^e = (p) */
2187 81717 : if (signe(m)) *pc = powii(p,m);
2188 81716 : if (!signe(r)) return mkvec2(gen_1,gen_0);
2189 40141 : q = p;
2190 40141 : gen = nfpow(nf, pr_get_gen(pr), r);
2191 : }
2192 : else
2193 : {
2194 114391 : m = absi_shallow(m);
2195 114391 : if (signe(r)) m = addiu(m,1);
2196 114391 : q = powii(p,m); /* m = ceil(|n|/e) */
2197 114390 : if (signe(n) >= 0) gen = nfpow(nf, pr_get_gen(pr), n);
2198 : else
2199 : {
2200 24029 : gen = pr_get_tau(pr);
2201 24029 : if (typ(gen) == t_MAT) gen = gel(gen,1);
2202 24029 : n = negi(n);
2203 24029 : gen = ZC_Z_divexact(nfpow(nf, gen, n), powii(p, subii(n,m)));
2204 24029 : *pc = ginv(q);
2205 : }
2206 : }
2207 154534 : gen = FpC_red(gen, q);
2208 : }
2209 690180 : return mkvec2(q, gen);
2210 : }
2211 :
2212 : /* True nf. x * pr^n. Assume x in HNF or scalar (possibly nonintegral) */
2213 : GEN
2214 575257 : idealmulpowprime(GEN nf, GEN x, GEN pr, GEN n)
2215 : {
2216 : GEN c, cx, y;
2217 575257 : long N = nf_get_degree(nf);
2218 :
2219 575302 : if (!signe(n)) return typ(x) == t_MAT? x: scalarmat_shallow(x, N);
2220 :
2221 : /* inert, special cased for efficiency */
2222 575295 : if (pr_is_inert(pr))
2223 : {
2224 64666 : GEN q = powii(pr_get_p(pr), n);
2225 62657 : return typ(x) == t_MAT? RgM_Rg_mul(x,q)
2226 127305 : : scalarmat_shallow(gmul(Q_abs(x),q), N);
2227 : }
2228 :
2229 510616 : y = idealpowprime(nf, pr, n, &c);
2230 510595 : if (typ(x) == t_MAT)
2231 508026 : { x = Q_primitive_part(x, &cx); if (is_pm1(gcoeff(x,1,1))) x = NULL; }
2232 : else
2233 2569 : { cx = x; x = NULL; }
2234 510518 : cx = mul_content(c,cx);
2235 510520 : if (x)
2236 349870 : x = idealHNF_mul_two(nf,x,y);
2237 : else
2238 160650 : x = idealhnf_two(nf,y);
2239 510663 : if (cx) x = ZM_Q_mul(x,cx);
2240 510358 : return x;
2241 : }
2242 : GEN
2243 5999 : idealdivpowprime(GEN nf, GEN x, GEN pr, GEN n)
2244 : {
2245 5999 : return idealmulpowprime(nf,x,pr, negi(n));
2246 : }
2247 :
2248 : /* nf = true nf */
2249 : static GEN
2250 788777 : idealpow_aux(GEN nf, GEN x, long tx, GEN n)
2251 : {
2252 788777 : GEN T = nf_get_pol(nf), m, cx, n1, a, alpha;
2253 788776 : long N = degpol(T), s = signe(n);
2254 788778 : if (!s) return matid(N);
2255 777179 : switch(tx)
2256 : {
2257 75747 : case id_PRINCIPAL:
2258 75747 : return idealhnf_principal(nf, nfpow(nf,x,n));
2259 505587 : case id_PRIME:
2260 505587 : if (pr_is_inert(x)) return scalarmat(powii(gel(x,1), n), N);
2261 419574 : x = idealpowprime(nf, x, n, &cx);
2262 419568 : x = idealhnf_two(nf,x);
2263 419581 : return cx? ZM_Q_mul(x, cx): x;
2264 195845 : default:
2265 195845 : if (is_pm1(n)) return (s < 0)? idealinv(nf, x): gcopy(x);
2266 71954 : n1 = (s < 0)? negi(n): n;
2267 :
2268 71954 : x = Q_primitive_part(x, &cx);
2269 71955 : a = mat_ideal_two_elt(nf,x); alpha = gel(a,2); a = gel(a,1);
2270 71955 : alpha = nfpow(nf,alpha,n1);
2271 71955 : m = zk_scalar_or_multable(nf, alpha);
2272 71955 : if (typ(m) == t_INT) {
2273 651 : x = gcdii(powii(a,n1), m);
2274 651 : if (s<0) x = ginv(x);
2275 651 : if (cx) x = gmul(x, powgi(cx,n));
2276 651 : x = scalarmat(x, N);
2277 : }
2278 : else
2279 : { /* could use gcdii(powii(a,n1), zkmultable_capZ(m)), but costly */
2280 71304 : x = ZM_hnfmodid(m, powii(a,n1));
2281 71304 : if (cx) cx = powgi(cx,n);
2282 71304 : if (s<0) {
2283 7 : GEN xZ = gcoeff(x,1,1);
2284 7 : cx = cx ? gdiv(cx, xZ): ginv(xZ);
2285 7 : x = idealHNF_inv_Z(nf,x);
2286 : }
2287 71304 : if (cx) x = ZM_Q_mul(x, cx);
2288 : }
2289 71955 : return x;
2290 : }
2291 : }
2292 :
2293 : /* raise the ideal x to the power n (in Z) */
2294 : GEN
2295 788778 : idealpow(GEN nf, GEN x, GEN n)
2296 : {
2297 : pari_sp av;
2298 : long tx;
2299 : GEN res, ax;
2300 :
2301 788778 : if (typ(n) != t_INT) pari_err_TYPE("idealpow",n);
2302 788778 : tx = idealtyp(&x,&ax);
2303 788780 : res = ax? cgetg(3,t_VEC): NULL;
2304 788778 : av = avma;
2305 788778 : x = gerepileupto(av, idealpow_aux(checknf(nf), x, tx, n));
2306 788772 : if (!ax) return x;
2307 0 : gel(res,1) = x;
2308 0 : gel(res,2) = ext_pow(nf, ax, n);
2309 0 : return res;
2310 : }
2311 :
2312 : /* Return ideal^e in number field nf. e is a C integer. */
2313 : GEN
2314 247908 : idealpows(GEN nf, GEN ideal, long e)
2315 : {
2316 247908 : long court[] = {evaltyp(t_INT) | _evallg(3),0,0};
2317 247908 : affsi(e,court); return idealpow(nf,ideal,court);
2318 : }
2319 :
2320 : static GEN
2321 28220 : _idealmulred(GEN nf, GEN x, GEN y)
2322 28220 : { return idealred(nf,idealmul(nf,x,y)); }
2323 : static GEN
2324 35821 : _idealsqrred(GEN nf, GEN x)
2325 35821 : { return idealred(nf,idealsqr(nf,x)); }
2326 : static GEN
2327 11350 : _mul(void *data, GEN x, GEN y) { return _idealmulred((GEN)data,x,y); }
2328 : static GEN
2329 35821 : _sqr(void *data, GEN x) { return _idealsqrred((GEN)data, x); }
2330 :
2331 : /* compute x^n (x ideal, n integer), reducing along the way */
2332 : GEN
2333 77215 : idealpowred(GEN nf, GEN x, GEN n)
2334 : {
2335 77215 : pari_sp av = avma, av2;
2336 : long s;
2337 : GEN y;
2338 :
2339 77215 : if (typ(n) != t_INT) pari_err_TYPE("idealpowred",n);
2340 77219 : s = signe(n); if (s == 0) return idealpow(nf,x,n);
2341 77219 : y = gen_pow_i(x, n, (void*)nf, &_sqr, &_mul);
2342 77219 : av2 = avma;
2343 77219 : if (s < 0) y = idealinv(nf,y);
2344 77219 : if (s < 0 || is_pm1(n)) y = idealred(nf,y);
2345 77220 : return avma == av2? gerepilecopy(av,y): gerepileupto(av,y);
2346 : }
2347 :
2348 : GEN
2349 16870 : idealmulred(GEN nf, GEN x, GEN y)
2350 : {
2351 16870 : pari_sp av = avma;
2352 16870 : return gerepileupto(av, _idealmulred(nf,x,y));
2353 : }
2354 :
2355 : long
2356 91 : isideal(GEN nf,GEN x)
2357 : {
2358 91 : long N, i, j, lx, tx = typ(x);
2359 : pari_sp av;
2360 : GEN T, xZ;
2361 :
2362 91 : nf = checknf(nf); T = nf_get_pol(nf); lx = lg(x);
2363 91 : if (tx==t_VEC && lx==3) { x = gel(x,1); tx = typ(x); lx = lg(x); }
2364 91 : switch(tx)
2365 : {
2366 14 : case t_INT: case t_FRAC: return 1;
2367 7 : case t_POL: return varn(x) == varn(T);
2368 7 : case t_POLMOD: return RgX_equal_var(T, gel(x,1));
2369 14 : case t_VEC: return get_prid(x)? 1 : 0;
2370 42 : case t_MAT: break;
2371 7 : default: return 0;
2372 : }
2373 42 : N = degpol(T);
2374 42 : if (lx-1 != N) return (lx == 1);
2375 28 : if (nbrows(x) != N) return 0;
2376 :
2377 28 : av = avma; x = Q_primpart(x);
2378 28 : if (!ZM_ishnf(x)) return 0;
2379 14 : xZ = gcoeff(x,1,1);
2380 21 : for (j=2; j<=N; j++)
2381 14 : if (!dvdii(xZ, gcoeff(x,j,j))) return gc_long(av,0);
2382 14 : for (i=2; i<=N; i++)
2383 14 : for (j=2; j<=N; j++)
2384 7 : if (! hnf_invimage(x, zk_ei_mul(nf,gel(x,i),j))) return gc_long(av,0);
2385 7 : return gc_long(av,1);
2386 : }
2387 :
2388 : GEN
2389 38717 : idealdiv(GEN nf, GEN x, GEN y)
2390 : {
2391 38717 : pari_sp av = avma, tetpil;
2392 38717 : GEN z = idealinv(nf,y);
2393 38717 : tetpil = avma; return gerepile(av,tetpil, idealmul(nf,x,z));
2394 : }
2395 :
2396 : /* This routine computes the quotient x/y of two ideals in the number field nf.
2397 : * It assumes that the quotient is an integral ideal. The idea is to find an
2398 : * ideal z dividing y such that gcd(Nx/Nz, Nz) = 1. Then
2399 : *
2400 : * x + (Nx/Nz) x
2401 : * ----------- = ---
2402 : * y + (Ny/Nz) y
2403 : *
2404 : * Proof: we can assume x and y are integral. Let p be any prime ideal
2405 : *
2406 : * If p | Nz, then it divides neither Nx/Nz nor Ny/Nz (since Nx/Nz is the
2407 : * product of the integers N(x/y) and N(y/z)). Both the numerator and the
2408 : * denominator on the left will be coprime to p. So will x/y, since x/y is
2409 : * assumed integral and its norm N(x/y) is coprime to p.
2410 : *
2411 : * If instead p does not divide Nz, then v_p (Nx/Nz) = v_p (Nx) >= v_p(x).
2412 : * Hence v_p (x + Nx/Nz) = v_p(x). Likewise for the denominators. QED.
2413 : *
2414 : * Peter Montgomery. July, 1994. */
2415 : static void
2416 7 : err_divexact(GEN x, GEN y)
2417 7 : { pari_err_DOMAIN("idealdivexact","denominator(x/y)", "!=",
2418 0 : gen_1,mkvec2(x,y)); }
2419 : GEN
2420 4050 : idealdivexact(GEN nf, GEN x0, GEN y0)
2421 : {
2422 4050 : pari_sp av = avma;
2423 : GEN x, y, xZ, yZ, Nx, Ny, Nz, cy, q, r;
2424 :
2425 4050 : nf = checknf(nf);
2426 4050 : x = idealhnf_shallow(nf, x0);
2427 4050 : y = idealhnf_shallow(nf, y0);
2428 4050 : if (lg(y) == 1) pari_err_INV("idealdivexact", y0);
2429 4043 : if (lg(x) == 1) { set_avma(av); return cgetg(1, t_MAT); } /* numerator is zero */
2430 4043 : y = Q_primitive_part(y, &cy);
2431 4043 : if (cy) x = RgM_Rg_div(x,cy);
2432 4043 : xZ = gcoeff(x,1,1); if (typ(xZ) != t_INT) err_divexact(x,y);
2433 4036 : yZ = gcoeff(y,1,1); if (isint1(yZ)) return gerepilecopy(av, x);
2434 2937 : Nx = idealnorm(nf,x);
2435 2937 : Ny = idealnorm(nf,y);
2436 2937 : if (typ(Nx) != t_INT) err_divexact(x,y);
2437 2937 : q = dvmdii(Nx,Ny, &r);
2438 2937 : if (signe(r)) err_divexact(x,y);
2439 2937 : if (is_pm1(q)) { set_avma(av); return matid(nf_get_degree(nf)); }
2440 : /* Find a norm Nz | Ny such that gcd(Nx/Nz, Nz) = 1 */
2441 599 : for (Nz = Ny;;) /* q = Nx/Nz */
2442 623 : {
2443 1222 : GEN p1 = gcdii(Nz, q);
2444 1222 : if (is_pm1(p1)) break;
2445 623 : Nz = diviiexact(Nz,p1);
2446 623 : q = mulii(q,p1);
2447 : }
2448 599 : xZ = gcoeff(x,1,1); q = gcdii(q, xZ);
2449 599 : if (!equalii(xZ,q))
2450 : { /* Replace x/y by x+(Nx/Nz) / y+(Ny/Nz) */
2451 354 : x = ZM_hnfmodid(x, q);
2452 : /* y reduced to unit ideal ? */
2453 354 : if (Nz == Ny) return gerepileupto(av, x);
2454 :
2455 140 : yZ = gcoeff(y,1,1); q = gcdii(diviiexact(Ny,Nz), yZ);
2456 140 : y = ZM_hnfmodid(y, q);
2457 : }
2458 385 : yZ = gcoeff(y,1,1);
2459 385 : y = idealHNF_mul(nf,x, idealHNF_inv_Z(nf,y));
2460 385 : return gerepileupto(av, ZM_Z_divexact(y, yZ));
2461 : }
2462 :
2463 : GEN
2464 21 : idealintersect(GEN nf, GEN x, GEN y)
2465 : {
2466 21 : pari_sp av = avma;
2467 : long lz, lx, i;
2468 : GEN z, dx, dy, xZ, yZ;;
2469 :
2470 21 : nf = checknf(nf);
2471 21 : x = idealhnf_shallow(nf,x);
2472 21 : y = idealhnf_shallow(nf,y);
2473 21 : if (lg(x) == 1 || lg(y) == 1) { set_avma(av); return cgetg(1,t_MAT); }
2474 14 : x = Q_remove_denom(x, &dx);
2475 14 : y = Q_remove_denom(y, &dy);
2476 14 : if (dx) y = ZM_Z_mul(y, dx);
2477 14 : if (dy) x = ZM_Z_mul(x, dy);
2478 14 : xZ = gcoeff(x,1,1);
2479 14 : yZ = gcoeff(y,1,1);
2480 14 : dx = mul_denom(dx,dy);
2481 14 : z = ZM_lll(shallowconcat(x,y), 0.99, LLL_KER); lz = lg(z);
2482 14 : lx = lg(x);
2483 63 : for (i=1; i<lz; i++) setlg(z[i], lx);
2484 14 : z = ZM_hnfmodid(ZM_mul(x,z), lcmii(xZ, yZ));
2485 14 : if (dx) z = RgM_Rg_div(z,dx);
2486 14 : return gerepileupto(av,z);
2487 : }
2488 :
2489 : /*******************************************************************/
2490 : /* */
2491 : /* T2-IDEAL REDUCTION */
2492 : /* */
2493 : /*******************************************************************/
2494 :
2495 : static GEN
2496 21 : chk_vdir(GEN nf, GEN vdir)
2497 : {
2498 21 : long i, l = lg(vdir);
2499 : GEN v;
2500 21 : if (l != lg(nf_get_roots(nf))) pari_err_DIM("idealred");
2501 14 : switch(typ(vdir))
2502 : {
2503 0 : case t_VECSMALL: return vdir;
2504 14 : case t_VEC: break;
2505 0 : default: pari_err_TYPE("idealred",vdir);
2506 : }
2507 14 : v = cgetg(l, t_VECSMALL);
2508 56 : for (i = 1; i < l; i++) v[i] = itos(gceil(gel(vdir,i)));
2509 14 : return v;
2510 : }
2511 :
2512 : static void
2513 12727 : twistG(GEN G, long r1, long i, long v)
2514 : {
2515 12727 : long j, lG = lg(G);
2516 12727 : if (i <= r1) {
2517 37321 : for (j=1; j<lG; j++) gcoeff(G,i,j) = gmul2n(gcoeff(G,i,j), v);
2518 : } else {
2519 651 : long k = (i<<1) - r1;
2520 4319 : for (j=1; j<lG; j++)
2521 : {
2522 3668 : gcoeff(G,k-1,j) = gmul2n(gcoeff(G,k-1,j), v);
2523 3668 : gcoeff(G,k ,j) = gmul2n(gcoeff(G,k ,j), v);
2524 : }
2525 : }
2526 12727 : }
2527 :
2528 : GEN
2529 135628 : nf_get_Gtwist(GEN nf, GEN vdir)
2530 : {
2531 : long i, l, v, r1;
2532 : GEN G;
2533 :
2534 135628 : if (!vdir) return nf_get_roundG(nf);
2535 21 : if (typ(vdir) == t_MAT)
2536 : {
2537 0 : long N = nf_get_degree(nf);
2538 0 : if (lg(vdir) != N+1 || lgcols(vdir) != N+1) pari_err_DIM("idealred");
2539 0 : return vdir;
2540 : }
2541 21 : vdir = chk_vdir(nf, vdir);
2542 14 : G = RgM_shallowcopy(nf_get_G(nf));
2543 14 : r1 = nf_get_r1(nf);
2544 14 : l = lg(vdir);
2545 56 : for (i=1; i<l; i++)
2546 : {
2547 42 : v = vdir[i]; if (!v) continue;
2548 42 : twistG(G, r1, i, v);
2549 : }
2550 14 : return RM_round_maxrank(G);
2551 : }
2552 : GEN
2553 12685 : nf_get_Gtwist1(GEN nf, long i)
2554 : {
2555 12685 : GEN G = RgM_shallowcopy( nf_get_G(nf) );
2556 12685 : long r1 = nf_get_r1(nf);
2557 12685 : twistG(G, r1, i, 10);
2558 12685 : return RM_round_maxrank(G);
2559 : }
2560 :
2561 : GEN
2562 95524 : RM_round_maxrank(GEN G0)
2563 : {
2564 95524 : long e, r = lg(G0)-1;
2565 95524 : pari_sp av = avma;
2566 95524 : for (e = 4; ; e <<= 1, set_avma(av))
2567 0 : {
2568 95524 : GEN G = gmul2n(G0, e), H = ground(G);
2569 95523 : if (ZM_rank(H) == r) return H; /* maximal rank ? */
2570 : }
2571 : }
2572 :
2573 : GEN
2574 135621 : idealred0(GEN nf, GEN I, GEN vdir)
2575 : {
2576 135621 : pari_sp av = avma;
2577 135621 : GEN G, aI, IZ, J, y, yZ, my, yi, c1 = NULL;
2578 : long N;
2579 :
2580 135621 : nf = checknf(nf);
2581 135621 : N = nf_get_degree(nf);
2582 : /* put first for sanity checks, unused when I obviously principal */
2583 135621 : G = nf_get_Gtwist(nf, vdir);
2584 135614 : switch (idealtyp(&I,&aI))
2585 : {
2586 34491 : case id_PRIME:
2587 34491 : if (pr_is_inert(I)) {
2588 585 : if (!aI) { set_avma(av); return matid(N); }
2589 585 : c1 = gel(I,1); I = matid(N);
2590 585 : goto END;
2591 : }
2592 33906 : IZ = pr_get_p(I);
2593 33906 : J = pr_inv_p(I);
2594 33907 : I = idealhnf_two(nf,I);
2595 33905 : break;
2596 101095 : case id_MAT:
2597 101095 : I = Q_primitive_part(I, &c1);
2598 101094 : IZ = gcoeff(I,1,1);
2599 101094 : if (is_pm1(IZ))
2600 : {
2601 8666 : if (!aI) { set_avma(av); return matid(N); }
2602 8631 : goto END;
2603 : }
2604 92428 : J = idealHNF_inv_Z(nf, I);
2605 92429 : break;
2606 21 : default: /* id_PRINCIPAL, silly case */
2607 21 : if (gequal0(I)) I = cgetg(1,t_MAT); else { c1 = I; I = matid(N); }
2608 21 : if (!aI) return I;
2609 14 : goto END;
2610 : }
2611 : /* now I integral, HNF; and J = (I\cap Z) I^(-1), integral */
2612 126334 : y = idealpseudomin(J, G); /* small elt in (I\cap Z)I^(-1), integral */
2613 126336 : if (equalii(ZV_content(y), IZ))
2614 : { /* already reduced */
2615 68688 : if (!aI) return gerepilecopy(av, I);
2616 65195 : goto END;
2617 : }
2618 :
2619 57648 : my = zk_multable(nf, y);
2620 57648 : I = ZM_Z_divexact(ZM_mul(my, I), IZ); /* y I / (I\cap Z), integral */
2621 57644 : c1 = mul_content(c1, IZ);
2622 57644 : if (equali1(c1)) c1 = NULL; /* can be simplified with IZ */
2623 57644 : yi = ZM_gauss(my, col_ei(N,1)); /* y^-1 */
2624 57648 : yZ = Q_denom(yi); /* (y) \cap Z */
2625 57647 : I = hnfmodid(I, yZ);
2626 57648 : if (!aI) return gerepileupto(av, I);
2627 55989 : if (typ(aI) == t_MAT) /* yi is not integral and usually larger than y */
2628 41436 : aI = famat_div(aI, y);
2629 14553 : else if (c1)
2630 14553 : c1 = RgC_Rg_mul(yi, c1);
2631 0 : END:
2632 130414 : if (c1) aI = ext_mul(nf, aI,c1);
2633 130413 : return gerepilecopy(av, mkvec2(I, aI));
2634 : }
2635 :
2636 : GEN
2637 7 : idealmin(GEN nf, GEN x, GEN vdir)
2638 : {
2639 7 : pari_sp av = avma;
2640 : GEN y, dx;
2641 7 : nf = checknf(nf);
2642 7 : switch( idealtyp(&x, NULL) )
2643 : {
2644 0 : case id_PRINCIPAL: return gcopy(x);
2645 0 : case id_PRIME: x = pr_hnf(nf,x); break;
2646 7 : case id_MAT: if (lg(x) == 1) return gen_0;
2647 : }
2648 7 : x = Q_remove_denom(x, &dx);
2649 7 : y = idealpseudomin(x, nf_get_Gtwist(nf,vdir));
2650 7 : if (dx) y = RgC_Rg_div(y, dx);
2651 7 : return gerepileupto(av, y);
2652 : }
2653 :
2654 : /*******************************************************************/
2655 : /* */
2656 : /* APPROXIMATION THEOREM */
2657 : /* */
2658 : /*******************************************************************/
2659 : /* a = ppi(a,b) ppo(a,b), where ppi regroups primes common to a and b
2660 : * and ppo(a,b) = Z_ppo(a,b) */
2661 : /* return gcd(a,b),ppi(a,b),ppo(a,b) */
2662 : GEN
2663 456155 : Z_ppio(GEN a, GEN b)
2664 : {
2665 456155 : GEN x, y, d = gcdii(a,b);
2666 456155 : if (is_pm1(d)) return mkvec3(gen_1, gen_1, a);
2667 346871 : x = d; y = diviiexact(a,d);
2668 : for(;;)
2669 63077 : {
2670 409948 : GEN g = gcdii(x,y);
2671 409948 : if (is_pm1(g)) return mkvec3(d, x, y);
2672 63077 : x = mulii(x,g); y = diviiexact(y,g);
2673 : }
2674 : }
2675 : /* a = ppg(a,b)pple(a,b), where ppg regroups primes such that v(a) > v(b)
2676 : * and pple all others */
2677 : /* return gcd(a,b),ppg(a,b),pple(a,b) */
2678 : GEN
2679 0 : Z_ppgle(GEN a, GEN b)
2680 : {
2681 0 : GEN x, y, g, d = gcdii(a,b);
2682 0 : if (equalii(a, d)) return mkvec3(a, gen_1, a);
2683 0 : x = diviiexact(a,d); y = d;
2684 : for(;;)
2685 : {
2686 0 : g = gcdii(x,y);
2687 0 : if (is_pm1(g)) return mkvec3(d, x, y);
2688 0 : x = mulii(x,g); y = diviiexact(y,g);
2689 : }
2690 : }
2691 : static void
2692 0 : Z_dcba_rec(GEN L, GEN a, GEN b)
2693 : {
2694 : GEN x, r, v, g, h, c, c0;
2695 : long n;
2696 0 : if (is_pm1(b)) {
2697 0 : if (!is_pm1(a)) vectrunc_append(L, a);
2698 0 : return;
2699 : }
2700 0 : v = Z_ppio(a,b);
2701 0 : a = gel(v,2);
2702 0 : r = gel(v,3);
2703 0 : if (!is_pm1(r)) vectrunc_append(L, r);
2704 0 : v = Z_ppgle(a,b);
2705 0 : g = gel(v,1);
2706 0 : h = gel(v,2);
2707 0 : x = c0 = gel(v,3);
2708 0 : for (n = 1; !is_pm1(h); n++)
2709 : {
2710 : GEN d, y;
2711 : long i;
2712 0 : v = Z_ppgle(h,sqri(g));
2713 0 : g = gel(v,1);
2714 0 : h = gel(v,2);
2715 0 : c = gel(v,3); if (is_pm1(c)) continue;
2716 0 : d = gcdii(c,b);
2717 0 : x = mulii(x,d);
2718 0 : y = d; for (i=1; i < n; i++) y = sqri(y);
2719 0 : Z_dcba_rec(L, diviiexact(c,y), d);
2720 : }
2721 0 : Z_dcba_rec(L,diviiexact(b,x), c0);
2722 : }
2723 : static GEN
2724 3088351 : Z_cba_rec(GEN L, GEN a, GEN b)
2725 : {
2726 : GEN g;
2727 3088351 : if (lg(L) > 10)
2728 : { /* a few naive steps before switching to dcba */
2729 0 : Z_dcba_rec(L, a, b);
2730 0 : return gel(L, lg(L)-1);
2731 : }
2732 3088351 : if (is_pm1(a)) return b;
2733 1835050 : g = gcdii(a,b);
2734 1835050 : if (is_pm1(g)) { vectrunc_append(L, a); return b; }
2735 1370894 : a = diviiexact(a,g);
2736 1370894 : b = diviiexact(b,g);
2737 1370894 : return Z_cba_rec(L, Z_cba_rec(L, a, g), b);
2738 : }
2739 : GEN
2740 346563 : Z_cba(GEN a, GEN b)
2741 : {
2742 346563 : GEN L = vectrunc_init(expi(a) + expi(b) + 2);
2743 346563 : GEN t = Z_cba_rec(L, a, b);
2744 346563 : if (!is_pm1(t)) vectrunc_append(L, t);
2745 346563 : return L;
2746 : }
2747 : /* P = coprime base, extend it by b; TODO: quadratic for now */
2748 : GEN
2749 0 : ZV_cba_extend(GEN P, GEN b)
2750 : {
2751 0 : long i, l = lg(P);
2752 0 : GEN w = cgetg(l+1, t_VEC);
2753 0 : for (i = 1; i < l; i++)
2754 : {
2755 0 : GEN v = Z_cba(gel(P,i), b);
2756 0 : long nv = lg(v)-1;
2757 0 : gel(w,i) = vecslice(v, 1, nv-1); /* those divide P[i] but not b */
2758 0 : b = gel(v,nv);
2759 : }
2760 0 : gel(w,l) = b; return shallowconcat1(w);
2761 : }
2762 : GEN
2763 0 : ZV_cba(GEN v)
2764 : {
2765 0 : long i, l = lg(v);
2766 : GEN P;
2767 0 : if (l <= 2) return v;
2768 0 : P = Z_cba(gel(v,1), gel(v,2));
2769 0 : for (i = 3; i < l; i++) P = ZV_cba_extend(P, gel(v,i));
2770 0 : return P;
2771 : }
2772 :
2773 : /* write x = x1 x2, x2 maximal s.t. (x2,f) = 1, return x2 */
2774 : GEN
2775 2232389 : Z_ppo(GEN x, GEN f)
2776 : {
2777 : for (;;)
2778 : {
2779 2232389 : f = gcdii(x, f); if (is_pm1(f)) break;
2780 1406715 : x = diviiexact(x, f);
2781 : }
2782 825671 : return x;
2783 : }
2784 : /* write x = x1 x2, x2 maximal s.t. (x2,f) = 1, return x2 */
2785 : ulong
2786 66767935 : u_ppo(ulong x, ulong f)
2787 : {
2788 : for (;;)
2789 : {
2790 66767935 : f = ugcd(x, f); if (f == 1) break;
2791 14899249 : x /= f;
2792 : }
2793 51868664 : return x;
2794 : }
2795 :
2796 : /* result known to be representable as an ulong */
2797 : static ulong
2798 2114552 : lcmuu(ulong a, ulong b) { ulong d = ugcd(a,b); return (a/d) * b; }
2799 :
2800 : /* assume 0 < x < N; return u in (Z/NZ)^* such that u x = gcd(x,N) (mod N);
2801 : * set *pd = gcd(x,N) */
2802 : ulong
2803 6495395 : Fl_invgen(ulong x, ulong N, ulong *pd)
2804 : {
2805 : ulong d, d0, e, v, v1;
2806 : long s;
2807 6495395 : *pd = d = xgcduu(N, x, 0, &v, &v1, &s);
2808 6495743 : if (s > 0) v = N - v;
2809 6495743 : if (d == 1) return v;
2810 : /* vx = gcd(x,N) (mod N), v coprime to N/d but need not be coprime to N */
2811 3348238 : e = N / d;
2812 3348238 : d0 = u_ppo(d, e); /* d = d0 d1, d0 coprime to N/d, rad(d1) | N/d */
2813 3348374 : if (d0 == 1) return v;
2814 2114504 : e = lcmuu(e, d / d0);
2815 2114542 : return u_chinese_coprime(v, 1, e, d0, e*d0);
2816 : }
2817 :
2818 : /* x t_INT, f ideal. Write x = x1 x2, sqf(x1) | f, (x2,f) = 1. Return x2 */
2819 : static GEN
2820 196 : nf_coprime_part(GEN nf, GEN x, GEN listpr)
2821 : {
2822 196 : long v, j, lp = lg(listpr), N = nf_get_degree(nf);
2823 : GEN x1, x2, ex;
2824 :
2825 : #if 0 /*1) via many gcds. Expensive ! */
2826 : GEN f = idealprodprime(nf, listpr);
2827 : f = ZM_hnfmodid(f, x); /* first gcd is less expensive since x in Z */
2828 : x = scalarmat(x, N);
2829 : for (;;)
2830 : {
2831 : if (gequal1(gcoeff(f,1,1))) break;
2832 : x = idealdivexact(nf, x, f);
2833 : f = ZM_hnfmodid(shallowconcat(f,x), gcoeff(x,1,1)); /* gcd(f,x) */
2834 : }
2835 : x2 = x;
2836 : #else /*2) from prime decomposition */
2837 196 : x1 = NULL;
2838 532 : for (j=1; j<lp; j++)
2839 : {
2840 336 : GEN pr = gel(listpr,j);
2841 336 : v = Z_pval(x, pr_get_p(pr)); if (!v) continue;
2842 :
2843 196 : ex = muluu(v, pr_get_e(pr)); /* = v_pr(x) > 0 */
2844 196 : x1 = x1? idealmulpowprime(nf, x1, pr, ex)
2845 196 : : idealpow(nf, pr, ex);
2846 : }
2847 196 : x = scalarmat(x, N);
2848 196 : x2 = x1? idealdivexact(nf, x, x1): x;
2849 : #endif
2850 196 : return x2;
2851 : }
2852 :
2853 : /* L0 in K^*, assume (L0,f) = 1. Return L integral, L0 = L mod f */
2854 : GEN
2855 10654 : make_integral(GEN nf, GEN L0, GEN f, GEN listpr)
2856 : {
2857 : GEN fZ, t, L, D2, d1, d2, d;
2858 :
2859 10654 : L = Q_remove_denom(L0, &d);
2860 10654 : if (!d) return L0;
2861 :
2862 : /* L0 = L / d, L integral */
2863 672 : fZ = gcoeff(f,1,1);
2864 672 : if (typ(L) == t_INT) return Fp_mul(L, Fp_inv(d, fZ), fZ);
2865 : /* Kill denom part coprime to fZ */
2866 196 : d2 = Z_ppo(d, fZ);
2867 196 : t = Fp_inv(d2, fZ); if (!is_pm1(t)) L = ZC_Z_mul(L,t);
2868 196 : if (equalii(d, d2)) return L;
2869 :
2870 196 : d1 = diviiexact(d, d2);
2871 : /* L0 = (L / d1) mod f. d1 not coprime to f
2872 : * write (d1) = D1 D2, D2 minimal, (D2,f) = 1. */
2873 196 : D2 = nf_coprime_part(nf, d1, listpr);
2874 196 : t = idealaddtoone_i(nf, D2, f); /* in D2, 1 mod f */
2875 196 : L = nfmuli(nf,t,L);
2876 :
2877 : /* if (L0, f) = 1, then L in D1 ==> in D1 D2 = (d1) */
2878 196 : return Q_div_to_int(L, d1); /* exact division */
2879 : }
2880 :
2881 : /* assume L is a list of prime ideals. Return the product */
2882 : GEN
2883 336 : idealprodprime(GEN nf, GEN L)
2884 : {
2885 336 : long l = lg(L), i;
2886 : GEN z;
2887 336 : if (l == 1) return matid(nf_get_degree(nf));
2888 336 : z = pr_hnf(nf, gel(L,1));
2889 371 : for (i=2; i<l; i++) z = idealHNF_mul_two(nf,z, gel(L,i));
2890 336 : return z;
2891 : }
2892 :
2893 : /* optimize for the frequent case I = nfhnf()[2]: lots of them are 1 */
2894 : GEN
2895 427 : idealprod(GEN nf, GEN I)
2896 : {
2897 427 : long i, l = lg(I);
2898 : GEN z;
2899 1043 : for (i = 1; i < l; i++)
2900 1036 : if (!equali1(gel(I,i))) break;
2901 427 : if (i == l) return gen_1;
2902 420 : z = gel(I,i);
2903 693 : for (i++; i<l; i++) z = idealmul(nf, z, gel(I,i));
2904 420 : return z;
2905 : }
2906 :
2907 : /* v_pr(idealprod(nf,I)) */
2908 : long
2909 2422 : idealprodval(GEN nf, GEN I, GEN pr)
2910 : {
2911 2422 : long i, l = lg(I), v = 0;
2912 14420 : for (i = 1; i < l; i++)
2913 11998 : if (!equali1(gel(I,i))) v += idealval(nf, gel(I,i), pr);
2914 2422 : return v;
2915 : }
2916 :
2917 : /* assume L is a list of prime ideals. Return prod L[i]^e[i] */
2918 : GEN
2919 55705 : factorbackprime(GEN nf, GEN L, GEN e)
2920 : {
2921 55705 : long l = lg(L), i;
2922 : GEN z;
2923 :
2924 55705 : if (l == 1) return matid(nf_get_degree(nf));
2925 42433 : z = idealpow(nf, gel(L,1), gel(e,1));
2926 65863 : for (i=2; i<l; i++)
2927 23429 : if (signe(gel(e,i))) z = idealmulpowprime(nf,z, gel(L,i),gel(e,i));
2928 42434 : return z;
2929 : }
2930 :
2931 : /* F in Z, divisible exactly by pr.p. Return F-uniformizer for pr, i.e.
2932 : * a t in Z_K such that v_pr(t) = 1 and (t, F/pr) = 1 */
2933 : GEN
2934 77234 : pr_uniformizer(GEN pr, GEN F)
2935 : {
2936 77234 : GEN p = pr_get_p(pr), t = pr_get_gen(pr);
2937 77234 : if (!equalii(F, p))
2938 : {
2939 43468 : long e = pr_get_e(pr);
2940 43468 : GEN u, v, q = (e == 1)? sqri(p): p;
2941 43468 : u = mulii(q, Fp_inv(q, diviiexact(F,p))); /* 1 mod F/p, 0 mod q */
2942 43467 : v = subui(1UL, u); /* 0 mod F/p, 1 mod q */
2943 43467 : if (pr_is_inert(pr))
2944 28 : t = addii(mulii(p, v), u);
2945 : else
2946 : {
2947 43439 : t = ZC_Z_mul(t, v);
2948 43440 : gel(t,1) = addii(gel(t,1), u); /* return u + vt */
2949 : }
2950 : }
2951 77234 : return t;
2952 : }
2953 : /* L = list of prime ideals, return lcm_i (L[i] \cap \ZM) */
2954 : GEN
2955 90923 : prV_lcm_capZ(GEN L)
2956 : {
2957 90923 : long i, r = lg(L);
2958 : GEN F;
2959 90923 : if (r == 1) return gen_1;
2960 78442 : F = pr_get_p(gel(L,1));
2961 141822 : for (i = 2; i < r; i++)
2962 : {
2963 63381 : GEN pr = gel(L,i), p = pr_get_p(pr);
2964 63381 : if (!dvdii(F, p)) F = mulii(F,p);
2965 : }
2966 78441 : return F;
2967 : }
2968 : /* v vector of prid. Return underlying list of rational primes */
2969 : GEN
2970 40005 : prV_primes(GEN v)
2971 : {
2972 40005 : long i, l = lg(v);
2973 40005 : GEN w = cgetg(l,t_VEC);
2974 110208 : for (i=1; i<l; i++) gel(w,i) = pr_get_p(gel(v,i));
2975 40005 : return ZV_sort_uniq(w);
2976 : }
2977 :
2978 : /* Given a prime ideal factorization with possibly zero or negative
2979 : * exponents, gives b such that v_p(b) = v_p(x) for all prime ideals pr | x
2980 : * and v_pr(b) >= 0 for all other pr.
2981 : * For optimal performance, all [anti-]uniformizers should be precomputed,
2982 : * but no support for this yet. If nored, do not reduce result. */
2983 : static GEN
2984 63308 : idealapprfact_i(GEN nf, GEN x, int nored)
2985 : {
2986 63308 : GEN d = NULL, z, L, e, e2, F;
2987 : long i, r;
2988 63308 : int hasden = 0;
2989 :
2990 63308 : nf = checknf(nf);
2991 63308 : L = gel(x,1);
2992 63308 : e = gel(x,2);
2993 63308 : F = prV_lcm_capZ(L);
2994 63306 : z = NULL; r = lg(e);
2995 165032 : for (i = 1; i < r; i++)
2996 : {
2997 101725 : long s = signe(gel(e,i));
2998 : GEN pi, q;
2999 101725 : if (!s) continue;
3000 72879 : if (s < 0) hasden = 1;
3001 72879 : pi = pr_uniformizer(gel(L,i), F);
3002 72880 : q = nfpow(nf, pi, gel(e,i));
3003 72880 : z = z? nfmul(nf, z, q): q;
3004 : }
3005 63307 : if (!z) return gen_1;
3006 36126 : if (hasden) /* denominator */
3007 : {
3008 10024 : z = Q_remove_denom(z, &d);
3009 10024 : d = diviiexact(d, Z_ppo(d, F));
3010 : }
3011 36126 : if (nored || typ(z) != t_COL) return d? gdiv(z, d): z;
3012 10023 : e2 = cgetg(r, t_VEC);
3013 28439 : for (i = 1; i < r; i++) gel(e2,i) = addiu(gel(e,i), 1);
3014 10022 : x = factorbackprime(nf, L, e2);
3015 10024 : if (d) x = RgM_Rg_mul(x, d);
3016 10023 : z = ZC_reducemodlll(z, x);
3017 10024 : return d? RgC_Rg_div(z,d): z;
3018 : }
3019 :
3020 : GEN
3021 0 : idealapprfact(GEN nf, GEN x) {
3022 0 : pari_sp av = avma;
3023 0 : return gerepileupto(av, idealapprfact_i(nf, x, 0));
3024 : }
3025 : GEN
3026 14 : idealappr(GEN nf, GEN x) {
3027 14 : pari_sp av = avma;
3028 14 : if (!is_nf_extfactor(x)) x = idealfactor(nf, x);
3029 14 : return gerepileupto(av, idealapprfact_i(nf, x, 0));
3030 : }
3031 :
3032 : /* OBSOLETE */
3033 : GEN
3034 14 : idealappr0(GEN nf, GEN x, long fl) { (void)fl; return idealappr(nf, x); }
3035 :
3036 : static GEN
3037 21 : mat_ideal_two_elt2(GEN nf, GEN x, GEN a)
3038 : {
3039 21 : GEN F = idealfactor(nf,a), P = gel(F,1), E = gel(F,2);
3040 21 : long i, r = lg(E);
3041 84 : for (i=1; i<r; i++) gel(E,i) = stoi( idealval(nf,x,gel(P,i)) );
3042 21 : return idealapprfact_i(nf,F,1);
3043 : }
3044 :
3045 : static void
3046 14 : not_in_ideal(GEN a) {
3047 14 : pari_err_DOMAIN("idealtwoelt2","element mod ideal", "!=", gen_0, a);
3048 0 : }
3049 : /* x integral in HNF, a an 'nf' */
3050 : static int
3051 28 : in_ideal(GEN x, GEN a)
3052 : {
3053 28 : switch(typ(a))
3054 : {
3055 14 : case t_INT: return dvdii(a, gcoeff(x,1,1));
3056 7 : case t_COL: return RgV_is_ZV(a) && !!hnf_invimage(x, a);
3057 7 : default: return 0;
3058 : }
3059 : }
3060 :
3061 : /* Given an integral ideal x and a in x, gives a b such that
3062 : * x = aZ_K + bZ_K using the approximation theorem */
3063 : GEN
3064 42 : idealtwoelt2(GEN nf, GEN x, GEN a)
3065 : {
3066 42 : pari_sp av = avma;
3067 : GEN cx, b;
3068 :
3069 42 : nf = checknf(nf);
3070 42 : a = nf_to_scalar_or_basis(nf, a);
3071 42 : x = idealhnf_shallow(nf,x);
3072 42 : if (lg(x) == 1)
3073 : {
3074 14 : if (!isintzero(a)) not_in_ideal(a);
3075 7 : set_avma(av); return gen_0;
3076 : }
3077 28 : x = Q_primitive_part(x, &cx);
3078 28 : if (cx) a = gdiv(a, cx);
3079 28 : if (!in_ideal(x, a)) not_in_ideal(a);
3080 21 : b = mat_ideal_two_elt2(nf, x, a);
3081 21 : if (typ(b) == t_COL)
3082 : {
3083 14 : GEN mod = idealhnf_principal(nf,a);
3084 14 : b = ZC_hnfrem(b,mod);
3085 14 : if (ZV_isscalar(b)) b = gel(b,1);
3086 : }
3087 : else
3088 : {
3089 7 : GEN aZ = typ(a) == t_COL? Q_denom(zk_inv(nf,a)): a; /* (a) \cap Z */
3090 7 : b = centermodii(b, aZ, shifti(aZ,-1));
3091 : }
3092 21 : b = cx? gmul(b,cx): gcopy(b);
3093 21 : return gerepileupto(av, b);
3094 : }
3095 :
3096 : /* Given 2 integral ideals x and y in nf, returns a beta in nf such that
3097 : * beta * x is an integral ideal coprime to y */
3098 : GEN
3099 37198 : idealcoprimefact(GEN nf, GEN x, GEN fy)
3100 : {
3101 37198 : GEN L = gel(fy,1), e;
3102 37198 : long i, r = lg(L);
3103 :
3104 37198 : e = cgetg(r, t_COL);
3105 76068 : for (i=1; i<r; i++) gel(e,i) = stoi( -idealval(nf,x,gel(L,i)) );
3106 37198 : return idealapprfact_i(nf, mkmat2(L,e), 0);
3107 : }
3108 : GEN
3109 84 : idealcoprime(GEN nf, GEN x, GEN y)
3110 : {
3111 84 : pari_sp av = avma;
3112 84 : return gerepileupto(av, idealcoprimefact(nf, x, idealfactor(nf,y)));
3113 : }
3114 :
3115 : GEN
3116 7 : nfmulmodpr(GEN nf, GEN x, GEN y, GEN modpr)
3117 : {
3118 7 : pari_sp av = avma;
3119 7 : GEN z, p, pr = modpr, T;
3120 :
3121 7 : nf = checknf(nf); modpr = nf_to_Fq_init(nf,&pr,&T,&p);
3122 0 : x = nf_to_Fq(nf,x,modpr);
3123 0 : y = nf_to_Fq(nf,y,modpr);
3124 0 : z = Fq_mul(x,y,T,p);
3125 0 : return gerepileupto(av, algtobasis(nf, Fq_to_nf(z,modpr)));
3126 : }
3127 :
3128 : GEN
3129 0 : nfdivmodpr(GEN nf, GEN x, GEN y, GEN modpr)
3130 : {
3131 0 : pari_sp av = avma;
3132 0 : nf = checknf(nf);
3133 0 : return gerepileupto(av, nfreducemodpr(nf, nfdiv(nf,x,y), modpr));
3134 : }
3135 :
3136 : GEN
3137 0 : nfpowmodpr(GEN nf, GEN x, GEN k, GEN modpr)
3138 : {
3139 0 : pari_sp av=avma;
3140 0 : GEN z, T, p, pr = modpr;
3141 :
3142 0 : nf = checknf(nf); modpr = nf_to_Fq_init(nf,&pr,&T,&p);
3143 0 : z = nf_to_Fq(nf,x,modpr);
3144 0 : z = Fq_pow(z,k,T,p);
3145 0 : return gerepileupto(av, algtobasis(nf, Fq_to_nf(z,modpr)));
3146 : }
3147 :
3148 : GEN
3149 0 : nfkermodpr(GEN nf, GEN x, GEN modpr)
3150 : {
3151 0 : pari_sp av = avma;
3152 0 : GEN T, p, pr = modpr;
3153 :
3154 0 : nf = checknf(nf); modpr = nf_to_Fq_init(nf, &pr,&T,&p);
3155 0 : if (typ(x)!=t_MAT) pari_err_TYPE("nfkermodpr",x);
3156 0 : x = nfM_to_FqM(x, nf, modpr);
3157 0 : return gerepilecopy(av, FqM_to_nfM(FqM_ker(x,T,p), modpr));
3158 : }
3159 :
3160 : GEN
3161 0 : nfsolvemodpr(GEN nf, GEN a, GEN b, GEN pr)
3162 : {
3163 0 : const char *f = "nfsolvemodpr";
3164 0 : pari_sp av = avma;
3165 : GEN T, p, modpr;
3166 :
3167 0 : nf = checknf(nf);
3168 0 : modpr = nf_to_Fq_init(nf, &pr,&T,&p);
3169 0 : if (typ(a)!=t_MAT) pari_err_TYPE(f,a);
3170 0 : a = nfM_to_FqM(a, nf, modpr);
3171 0 : switch(typ(b))
3172 : {
3173 0 : case t_MAT:
3174 0 : b = nfM_to_FqM(b, nf, modpr);
3175 0 : b = FqM_gauss(a,b,T,p);
3176 0 : if (!b) pari_err_INV(f,a);
3177 0 : a = FqM_to_nfM(b, modpr);
3178 0 : break;
3179 0 : case t_COL:
3180 0 : b = nfV_to_FqV(b, nf, modpr);
3181 0 : b = FqM_FqC_gauss(a,b,T,p);
3182 0 : if (!b) pari_err_INV(f,a);
3183 0 : a = FqV_to_nfV(b, modpr);
3184 0 : break;
3185 0 : default: pari_err_TYPE(f,b);
3186 : }
3187 0 : return gerepilecopy(av, a);
3188 : }
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