# HG changeset patch # User Jeroen Demeyer # Date 1282168698 -7200 # Node ID cd0623e7596fe89d62fe973fb649b2e50c3530fd # Parent cd88cc8db5eeb3d9da9c0378669c922fe9ca70f9 #9764: implement hashing for number field ideals that uses the HNF form instead of a string #9764: make number field ideals *not* print in reduced form, use gens_two() if the discriminant is large #9764: Use PARI's hash_GEN() for gen.__hash__ diff -r cd88cc8db5ee -r cd0623e7596f sage/libs/pari/gen.pyx --- a/sage/libs/pari/gen.pyx Thu Sep 16 02:43:11 2010 +0200 +++ b/sage/libs/pari/gen.pyx Wed Aug 18 23:58:18 2010 +0200 @@ -416,8 +416,19 @@ return P.new_gen_to_string(self.g) def __hash__(self): - _sig_on - return hash(P.new_gen_to_string(self.g)) + """ + Return the hash of self, computed using PARI's hash_GEN(). + + TESTS:: + + sage: type(pari('1 + 2.0*I').__hash__()) + + """ + cdef long h + _sig_on + h = hash_GEN(self.g) + _sig_off + return h def _testclass(self): import test diff -r cd88cc8db5ee -r cd0623e7596f sage/rings/number_field/number_field.py --- a/sage/rings/number_field/number_field.py Thu Sep 16 02:43:11 2010 +0200 +++ b/sage/rings/number_field/number_field.py Wed Aug 18 23:58:18 2010 +0200 @@ -4285,7 +4285,7 @@ sage: K. = NumberField(x^4+3*x^2-17) sage: P = K.ideal(61).factor()[0][0] sage: K.residue_field(P) - Residue field in abar of Fractional ideal (-2*a^2 + 1) + Residue field in abar of Fractional ideal (61, a^2 + 30) :: @@ -4299,7 +4299,7 @@ sage: L.residue_field(P) Traceback (most recent call last): ... - ValueError: Fractional ideal (-2*a^2 + 1) is not an ideal of Number Field in b with defining polynomial x^2 + 5 + ValueError: Fractional ideal (61, a^2 + 30) is not an ideal of Number Field in b with defining polynomial x^2 + 5 sage: L.residue_field(2) Traceback (most recent call last): ... @@ -7043,7 +7043,7 @@ sage: C20 = CyclotomicField(20) sage: C20.different() - Fractional ideal (-4*zeta20^7 + 8*zeta20^5 - 12*zeta20^3 + 6*zeta20) + Fractional ideal (10, 2*zeta20^6 - 4*zeta20^4 - 4*zeta20^2 + 2) sage: C18 = CyclotomicField(18) sage: D = C18.different().norm() sage: D == C18.discriminant().abs() diff -r cd88cc8db5ee -r cd0623e7596f sage/rings/number_field/number_field_ideal.py --- a/sage/rings/number_field/number_field_ideal.py Thu Sep 16 02:43:11 2010 +0200 +++ b/sage/rings/number_field/number_field_ideal.py Wed Aug 18 23:58:18 2010 +0200 @@ -36,6 +36,8 @@ # http://www.gnu.org/licenses/ #***************************************************************************** +SMALL_DISC = 1000000 + import operator import sage.misc.latex as latex @@ -178,6 +180,21 @@ if len(gens)==0: raise ValueError, "gens must have length at least 1 (zero ideal is not a fractional ideal)" Ideal_generic.__init__(self, field, gens, coerce) + + def __hash__(self): + """ + EXAMPLES:: + + sage: NumberField(x^2 + 1, 'a').ideal(7).__hash__() + -9223372036854775779 # 64-bit + -2147483619 # 32-bit + """ + try: return self._hash + # At some point in the future (e.g., for relative extensions), we'll likely + # have to consider other hashes, like the following. + #except AttributeError: self._hash = hash(self.gens()) + except AttributeError: self._hash = self.pari_hnf().__hash__() + return self._hash def _latex_(self): """ @@ -188,8 +205,27 @@ '\\left(2, \\frac{1}{2} a - \\frac{1}{2}\\right)' sage: latex(K.ideal([2, 1/2*a - 1/2])) \left(2, \frac{1}{2} a - \frac{1}{2}\right) + + The gens are reduced only if the norm of the discriminant of + the defining polynomial is at most + sage.rings.number_field.number_field_ideal.SMALL_DISC:: + + sage: K. = NumberField(x^2 + 902384092834); K + Number Field in a with defining polynomial x^2 + 902384092834 + sage: I = K.factor(19)[0][0]; I._latex_() + '\\left(19\\right)' + + We can make the generators reduced by increasing SMALL_DISC. + We had better also set proof to False, or computing reduced + gens could take too long:: + + sage: proof.number_field(False) + sage: sage.rings.number_field.number_field_ideal.SMALL_DISC = 10^20 + sage: K. = NumberField(x^4 + 3*x^2 - 17) + sage: K.ideal([17*a,17,17,17*a])._latex_() + '\\left(17\\right)' """ - return '\\left(%s\\right)'%(", ".join(map(latex.latex, self.gens_reduced()))) + return '\\left(%s\\right)'%(", ".join(map(latex.latex, self._gens_repr()))) def __cmp__(self, other): """ @@ -363,7 +399,14 @@ def _repr_short(self): """ - Efficient string representation of this fraction ideal. + Compact string representation of this ideal. When the norm of + the discriminant of the defining polynomial of the number field + is less than + + sage.rings.number_field.number_field_ideal.SMALL_DISC + + then display reduced generators. Otherwise display two + generators. EXAMPLES:: @@ -372,13 +415,63 @@ sage: I = K.factor(17)[0][0]; I Fractional ideal (17, a^2 - 6) sage: I._repr_short() - '(17, a^2 - 6)' + '(17, a^2 - 6)' + + We use reduced gens, because the discriminant is small:: + + sage: K. = NumberField(x^2 + 17); K + Number Field in a with defining polynomial x^2 + 17 + sage: I = K.factor(17)[0][0]; I + Fractional ideal (-a) + + Here the discriminant is 'large', so the gens aren't reduced:: + + sage: sage.rings.number_field.number_field_ideal.SMALL_DISC + 1000000 + sage: K. = NumberField(x^2 + 902384094); K + Number Field in a with defining polynomial x^2 + 902384094 + sage: I = K.factor(19)[0][0]; I + Fractional ideal (19, a + 14) + sage: I.gens_reduced() # long time + (19, a + 14) """ - #NOTE -- we will *have* to not reduce the gens soon, since this - # makes things insanely slow in general. - # When I fix this, I *have* to also change the _latex_ method. - return '(%s)'%(', '.join(map(str, self.gens_reduced()))) -# return '(%s)'%(', '.join(map(str, self.gens()))) + return '(%s)'%(', '.join(map(str, self._gens_repr()))) + + def _gens_repr(self): + """ + Returns tuple of generators to be used for printing this number + field ideal. The gens are reduced only if the absolute value of + the norm of the discriminant of the defining polynomial is at + most sage.rings.number_field.number_field_ideal.SMALL_DISC. + + EXAMPLES:: + + sage: sage.rings.number_field.number_field_ideal.SMALL_DISC + 1000000 + sage: K. = NumberField(x^4 + 3*x^2 - 17) + sage: K.discriminant() # too big + -1612688 + sage: I = K.ideal([17*a*(2*a-2),17*a*(2*a-3)]); I._gens_repr() + (289, 17*a) + sage: I.gens_reduced() + (17*a,) + """ + # If the discriminant is small, it is easy to find nice gens. + # Otherwise it is potentially very hard. + try: + if abs(self.number_field().defining_polynomial().discriminant().norm()) <= SMALL_DISC: + return self.gens_reduced() + except TypeError: + # In some cases with relative extensions, computing the + # discriminant of the defining polynomial is not + # supported. + pass + # Return two generators unless the second one is zero + two_gens = self.gens_two() + if two_gens[1]: + return two_gens + else: + return (two_gens[0],) def _pari_(self): """ @@ -604,18 +697,21 @@ sage: J = K.ideal([a+2, 9]) sage: J.gens() (a + 2, 9) - sage: J.gens_reduced() + sage: J.gens_reduced() # random sign (a + 2,) sage: K.ideal([a+2, 3]).gens_reduced() (3, a + 2) TESTS:: + sage: len(J.gens_reduced()) == 1 + True + sage: all(j.parent() is K for j in J.gens()) True sage: all(j.parent() is K for j in J.gens_reduced()) True - + sage: K. = NumberField(x^4 + 10*x^2 + 20) sage: J = K.prime_above(5) sage: J.is_principal() @@ -1328,7 +1424,7 @@ sage: K. = CyclotomicField(11); K Cyclotomic Field of order 11 and degree 10 sage: I = K.factor(31)[0][0]; I - Fractional ideal (-3*a^7 - 4*a^5 - 3*a^4 - 3*a^2 - 3*a - 3) + Fractional ideal (31, a^5 + 10*a^4 - a^3 + a^2 + 9*a - 1) sage: I.divides(I) True sage: I.divides(31) @@ -1351,11 +1447,11 @@ sage: I = K.ideal(19); I Fractional ideal (19) sage: F = I.factor(); F - (Fractional ideal (a^2 + 2*a + 2)) * (Fractional ideal (a^2 - 2*a + 2)) + (Fractional ideal (19, 1/2*a^2 + a - 17/2)) * (Fractional ideal (19, 1/2*a^2 - a - 17/2)) sage: type(F) sage: list(F) - [(Fractional ideal (a^2 + 2*a + 2), 1), (Fractional ideal (a^2 - 2*a + 2), 1)] + [(Fractional ideal (19, 1/2*a^2 + a - 17/2), 1), (Fractional ideal (19, 1/2*a^2 - a - 17/2), 1)] sage: F.prod() Fractional ideal (19) """ diff -r cd88cc8db5ee -r cd0623e7596f sage/rings/number_field/number_field_ideal_rel.py --- a/sage/rings/number_field/number_field_ideal_rel.py Thu Sep 16 02:43:11 2010 +0200 +++ b/sage/rings/number_field/number_field_ideal_rel.py Wed Aug 18 23:58:18 2010 +0200 @@ -192,7 +192,8 @@ sage: J.absolute_norm() 22584817 sage: J.absolute_ideal() - Fractional ideal (188/812911*a^5 - 1/812911*a^4 + 45120/812911*a^3 - 56/73901*a^2 + 3881638/812911*a + 50041/812911) + Fractional ideal (22584817, -4417/2438733*a^5 - 2867/1625822*a^4 - 1307249/4877466*a^3 - 112151/443406*a^2 - 22904674/2438733*a - 13720435234111/2438733) # 32-bit + Fractional ideal (22584817, -1473/812911*a^5 + 8695/4877466*a^4 - 1308209/4877466*a^3 + 117415/443406*a^2 - 22963264/2438733*a - 13721081784272/2438733) # 64-bit sage: J.absolute_ideal().norm() 22584817 @@ -414,8 +415,7 @@ sage: K. = F.extension(Y^2 - (1 + a)*(a + b)*a*b) sage: I = K.ideal(3, c) sage: J = I.ideal_below(); J - Fractional ideal (b) # 32-bit - Fractional ideal (-b) # 64-bit + Fractional ideal (b) sage: J.number_field() == F True """ @@ -593,8 +593,8 @@ sage: I.relative_ramification_index() 2 sage: I.ideal_below() - Fractional ideal (b) # 32-bit - Fractional ideal (-b) # 64-bit + Fractional ideal (-b) # 32-bit + Fractional ideal (b) # 64-bit sage: K.ideal(b) == I^2 True """ @@ -770,7 +770,7 @@ sage: K. = NumberField(x^2+6) sage: L. = K.extension(K['x'].gen()^4 + a) sage: I = L.ideal(b); I - Fractional ideal (b) + Fractional ideal (6, b) sage: is_NumberFieldFractionalIdeal_rel(I) True sage: N = I.relative_norm(); N diff -r cd88cc8db5ee -r cd0623e7596f sage/rings/number_field/order.py --- a/sage/rings/number_field/order.py Thu Sep 16 02:43:11 2010 +0200 +++ b/sage/rings/number_field/order.py Wed Aug 18 23:58:18 2010 +0200 @@ -673,7 +673,7 @@ sage: P = K.ideal(61).factor()[0][0] sage: OK = K.maximal_order() sage: OK.residue_field(P) - Residue field in abar of Fractional ideal (-2*a^2 + 1) + Residue field in abar of Fractional ideal (61, a^2 + 30) """ if self.is_maximal(): return self.number_field().residue_field(prime, name, check) diff -r cd88cc8db5ee -r cd0623e7596f sage/rings/number_field/small_primes_of_degree_one.py --- a/sage/rings/number_field/small_primes_of_degree_one.py Thu Sep 16 02:43:11 2010 +0200 +++ b/sage/rings/number_field/small_primes_of_degree_one.py Wed Aug 18 23:58:18 2010 +0200 @@ -46,7 +46,7 @@ EXAMPLES:: sage: x = ZZ['x'].gen() - sage: F. = NumberField(x^2 - 2, 'a') + sage: F. = NumberField(x^2 - 2) sage: Ps = F.primes_of_degree_one_list(3) sage: Ps # random [Fractional ideal (2*a + 1), Fractional ideal (-3*a + 1), Fractional ideal (-a + 5)] @@ -59,7 +59,7 @@ The next two examples are for relative number fields.:: - sage: L. = F.extension(x^3 - a, 'b') + sage: L. = F.extension(x^3 - a) sage: Ps = L.primes_of_degree_one_list(3) sage: Ps # random [Fractional ideal (17, b - 5), Fractional ideal (23, b - 4), Fractional ideal (31, b - 2)] @@ -69,7 +69,7 @@ True sage: all(P.residue_class_degree() == 1 for P in Ps) True - sage: M. = NumberField(x^2 - x*b^2 + b, 'c') + sage: M. = NumberField(x^2 - x*b^2 + b) sage: Ps = M.primes_of_degree_one_list(3) sage: Ps # random [Fractional ideal (17, c - 2), Fractional ideal (c - 1), Fractional ideal (41, c + 15)] diff -r cd88cc8db5ee -r cd0623e7596f sage/rings/polynomial/polynomial_quotient_ring.py --- a/sage/rings/polynomial/polynomial_quotient_ring.py Thu Sep 16 02:43:11 2010 +0200 +++ b/sage/rings/polynomial/polynomial_quotient_ring.py Wed Aug 18 23:58:18 2010 +0200 @@ -689,8 +689,8 @@ `x^2 + 31` from 12 to 2, i.e. we lose a generator of order 6:: sage: S.S_class_group([K.ideal(a)]) - [((1/4*xbar^2 + 31/4, (-1/8*a + 1/8)*xbar^2 - 31/8*a + 31/8, 1/16*xbar^3 + 1/16*xbar^2 + 31/16*xbar + 31/16, -1/16*a*xbar^3 + (1/16*a + 1/8)*xbar^2 - 31/16*a*xbar + 31/16*a + 31/8), 6, 1/16*xbar^3 - 5/16*xbar^2 + 31/16*xbar - 139/16), ((-1/4*xbar^2 - 23/4, (1/8*a - 1/8)*xbar^2 + 23/8*a - 23/8, -1/16*xbar^3 - 1/16*xbar^2 - 23/16*xbar - 23/16, 1/16*a*xbar^3 + (-1/16*a - 1/8)*xbar^2 + 23/16*a*xbar - 23/16*a - 23/8), 2, -1/8*xbar^2 - 15/8)] # 32-bit - [((1/4*xbar^2 + 31/4, (-1/8*a + 1/8)*xbar^2 - 31/8*a + 31/8, 1/16*xbar^3 + 1/16*xbar^2 + 31/16*xbar + 31/16, -1/16*a*xbar^3 + (1/16*a + 1/8)*xbar^2 - 31/16*a*xbar + 31/16*a + 31/8), 6, 1/16*xbar^3 - 5/16*xbar^2 + 31/16*xbar - 139/16), ((-1/4*xbar^2 - 23/4, (1/8*a - 1/8)*xbar^2 + 23/8*a - 23/8, -1/16*xbar^3 - 1/16*xbar^2 - 23/16*xbar - 23/16, 1/16*a*xbar^3 + (-1/16*a - 1/8)*xbar^2 + 23/16*a*xbar - 23/16*a - 23/8), 2, -1/8*xbar^2 - 15/8)] # 64-bit + [((1/4*xbar^2 + 31/4, (-1/8*a + 1/8)*xbar^2 - 31/8*a + 31/8, 1/16*xbar^3 + 1/16*xbar^2 + 31/16*xbar + 31/16, -1/16*a*xbar^3 + (1/16*a + 1/8)*xbar^2 - 31/16*a*xbar + 31/16*a + 31/8), 6, 1/16*xbar^3 - 5/16*xbar^2 + 31/16*xbar - 139/16), ((-1/4*xbar^2 - 23/4, (1/8*a - 1/8)*xbar^2 + 23/8*a - 23/8, -1/16*xbar^3 - 1/16*xbar^2 - 23/16*xbar - 23/16, 1/16*a*xbar^3 + (-1/16*a - 1/8)*xbar^2 + 23/16*a*xbar - 23/16*a - 23/8), 2, -1/8*xbar^2 - 15/8)] # 32-bit + [((1/4*xbar^2 + 31/4, (-1/8*a + 1/8)*xbar^2 - 31/8*a + 31/8, 1/16*xbar^3 + 1/16*xbar^2 + 31/16*xbar + 31/16, -1/16*a*xbar^3 + (1/16*a + 1/8)*xbar^2 - 31/16*a*xbar + 31/16*a + 31/8), 6, -1/16*xbar^3 + 1/16*xbar^2 - 31/16*xbar + 47/16), ((-1/4*xbar^2 - 23/4, (1/8*a - 1/8)*xbar^2 + 23/8*a - 23/8, -1/16*xbar^3 - 1/16*xbar^2 - 23/16*xbar - 23/16, 1/16*a*xbar^3 + (-1/16*a - 1/8)*xbar^2 + 23/16*a*xbar - 23/16*a - 23/8), 2, -1/8*xbar^2 - 15/8)] # 64-bit Note that all the returned values live where we expect them to:: diff -r cd88cc8db5ee -r cd0623e7596f sage/rings/residue_field.pyx --- a/sage/rings/residue_field.pyx Thu Sep 16 02:43:11 2010 +0200 +++ b/sage/rings/residue_field.pyx Wed Aug 18 23:58:18 2010 +0200 @@ -172,7 +172,7 @@ An example where the residue class field is large but of degree 1:: sage: K. = NumberField(x^3-875); P = K.ideal(2007).factor()[0][0]; k = K.residue_field(P); k - Residue field of Fractional ideal (-2/25*a^2 - 2/5*a - 3) + Residue field of Fractional ideal (223, 1/5*a + 11) sage: k(a) 168 sage: k(a)^3 - 875 @@ -468,8 +468,7 @@ sage: K. = NumberField(x^3-11) sage: F = K.ideal(37).factor(); F - (Fractional ideal (37, a + 12)) * (Fractional ideal (2*a - 5)) * (Fractional ideal (37, a + 9)) # 32-bit - (Fractional ideal (37, a + 12)) * (Fractional ideal (-2*a + 5)) * (Fractional ideal (37, a + 9)) # 64-bit + (Fractional ideal (37, a + 12)) * (Fractional ideal (-2*a + 5)) * (Fractional ideal (37, a + 9)) sage: k = K.residue_field(F[0][0]) sage: l = K.residue_field(F[1][0]) sage: k == l