# HG changeset patch # User John Cremona # Date 1303483826 -3600 # Node ID 2053d9b1f2723b8b9ef48103582074c31584b0f9 # Parent 9418a12e694afd8af736d4892d1a66a7d6145641 #11220: implement CM j-invariants and orders for quadratic fields diff -r 9418a12e694a -r 2053d9b1f272 sage/schemes/elliptic_curves/cm.py --- a/sage/schemes/elliptic_curves/cm.py Tue Apr 19 18:16:07 2011 +0000 +++ b/sage/schemes/elliptic_curves/cm.py Fri Apr 22 15:50:26 2011 +0100 @@ -150,7 +150,7 @@ INPUT: - - ``K`` -- a number field (currently only implemented for `K=\QQ`) + - ``K`` -- a number field (currently only implemented for `K` of degree at most 2) OUTPUT: @@ -158,21 +158,31 @@ .. note:: - This is currently only implemented for the rationals. David - Kohel has large tables for other fields, but they are not in - Sage yet. + This is currently only implemented for the rationals and + quadratic fields. David Kohel has large tables for other + functions, but they are not in Sage yet. EXAMPLE:: sage: cm_j_invariants(QQ) [0, 54000, -12288000, 1728, 287496, -3375, 16581375, 8000, -32768, -884736, -884736000, -147197952000, -262537412640768000] - :: + Over imaginary quadratic fields there are no more than over `QQ`:: - sage: cm_j_invariants(QuadraticField(-1, 'i')) + sage: cm_j_invariants(QuadraticField(-1, 'i')) + [0, 54000, -12288000, 1728, 287496, -3375, 16581375, 8000, -32768, -884736, -884736000, -147197952000, -262537412640768000] + + Over real quadratic fields there may be more:: + + sage: cm_j_invariants(QuadraticField(5, 'a')) + [0, 54000, -12288000, 1728, 287496, -3375, 16581375, 8000, -32768, -884736, -884736000, -147197952000, -262537412640768000, 282880*a + 632000, -282880*a + 632000, 95178240*a + 212846400, -95178240*a + 212846400, 85995/2*a - 191025/2, -85995/2*a - 191025/2, 26378240*a - 58982400, -26378240*a - 58982400, 95673435586560*a - 213932305612800, -95673435586560*a - 213932305612800, 184068066743177379840*a - 411588709724712960000, -184068066743177379840*a - 411588709724712960000, 16554983445/2*a + 37018076625/2, -16554983445/2*a + 37018076625/2] + + Over fields of higher degree this is not yet implemented:: + + sage: K. = NumberField(x^3-2) + sage: cm_j_invariants(K) Traceback (most recent call last): - ... - NotImplementedError: Enumeration of CM j-invariants over Number Field in i with defining polynomial x^2 + 1 not yet implemented + NotImplementedError: Enumeration of CM j-invariants over fields of degree 3 not yet implemented """ if K == RationalField(): @@ -180,8 +190,14 @@ 287496, -3375, 16581375, 8000, \ -32768, -884736, -884736000,\ -147197952000, -262537412640768000]] - else: - raise NotImplementedError, "Enumeration of CM j-invariants over %s not yet implemented"%K + if K.degree()==2: + # list the discriminants of class number 2: + d2list = [-5,-6,-10,-13,-15,-22,-35,-37,-51,-58,-91,-115,-123,-187,-235,-267,-403,-427] + d2list = [d if d%4==1 else 4*d for d in d2list] + d2list = d2list + [-60] # the only non-maximal order of class number 2 + return sum([hilbert_class_polynomial(D).roots(K,multiplicities=False) for D in d2list],cm_j_invariants(RationalField())) + + raise NotImplementedError, "Enumeration of CM j-invariants over fields of degree %s not yet implemented"%K.degree() def cm_j_invariants_and_orders(K): @@ -190,25 +206,36 @@ INPUT: - - ``K`` -- a number field (currently only implemented for `K=\QQ`) + - ``K`` -- a number field (currently only implemented for `K` of degree at most 2) OUTPUT: (list) A list of 3-tuples `(D,f,j)` where `j` is a CM - `j`-invariant with quadratic fundamental discriminant `D` and - conductor `f` + `j`-invariant in `K` with quadratic fundamental discriminant `D` + and conductor `f`. EXAMPLE:: sage: cm_j_invariants_and_orders(QQ) [(-3, 3, -12288000), (-3, 2, 54000), (-3, 1, 0), (-4, 2, 287496), (-4, 1, 1728), (-7, 2, 16581375), (-7, 1, -3375), (-8, 1, 8000), (-11, 1, -32768), (-19, 1, -884736), (-43, 1, -884736000), (-67, 1, -147197952000), (-163, 1, -262537412640768000)] - :: + Over an imaginary quadratic field there are no more than over `QQ`:: + + sage: cm_j_invariants_and_orders(QuadraticField(-1, 'i')) + [(-3, 3, -12288000), (-3, 2, 54000), (-3, 1, 0), (-4, 2, 287496), (-4, 1, 1728), (-7, 2, 16581375), (-7, 1, -3375), (-8, 1, 8000), (-11, 1, -32768), (-19, 1, -884736), (-43, 1, -884736000), (-67, 1, -147197952000), (-163, 1, -262537412640768000)] + + Over real quadratic fields there may be more:: + + sage: cm_j_invariants_and_orders(QuadraticField(5,'a')) + [(-20, 1, 282880*a + 632000), (-24, 1, -282880*a + 632000), (-40, 1, 95178240*a + 212846400), (-52, 1, -95178240*a + 212846400), (-15, 1, 85995/2*a - 191025/2), (-88, 1, -85995/2*a - 191025/2), (-35, 1, 26378240*a - 58982400), (-148, 1, -26378240*a - 58982400), (-51, 1, 95673435586560*a - 213932305612800), (-232, 1, -95673435586560*a - 213932305612800), (-91, 1, 184068066743177379840*a - 411588709724712960000), (-115, 1, -184068066743177379840*a - 411588709724712960000), (-123, 1, 16554983445/2*a + 37018076625/2), (-187, 1, -16554983445/2*a + 37018076625/2), (-3, 3, -12288000), (-3, 2, 54000), (-3, 1, 0), (-4, 2, 287496), (-4, 1, 1728), (-7, 2, 16581375), (-7, 1, -3375), (-8, 1, 8000), (-11, 1, -32768), (-19, 1, -884736), (-43, 1, -884736000), (-67, 1, -147197952000), (-163, 1, -262537412640768000)] + + Over fields of higher degree this is not yet implemented:: - sage: cm_j_invariants_and_orders(QuadraticField(-1, 'i')) + sage: K. = NumberField(x^3-2) + sage: cm_j_invariants_and_orders(K) Traceback (most recent call last): - ... - NotImplementedError: Enumeration of CM j-invariants over Number Field in i with defining polynomial x^2 + 1 not yet implemented + NotImplementedError: Enumeration of CM j-invariants over fields of degree 3 not yet implemented + """ if K == RationalField(): @@ -222,6 +249,15 @@ T.sort() T.reverse() return T - else: - raise NotImplementedError, "Enumeration of CM j-invariants over %s not yet implemented"%K + if K.degree()==2: + # list the discriminants of class number 2: + dlist = [-5,-6,-10,-13,-15,-22,-35,-37,-51,-58,-91,-115,-123,-187,-235,-267,-403,-427] + dlist = [d if d%4==1 else 4*d for d in dlist] + d2list = dlist + [-60] # the only non-maximal order of class number 2 + D2list = dlist + [-15] + flist = [1 for d in dlist] + [2] + return zip(D2list,flist,sum([hilbert_class_polynomial(D).roots(K,multiplicities=False) for D in d2list],[])) + cm_j_invariants_and_orders(RationalField()) + + raise NotImplementedError, "Enumeration of CM j-invariants over fields of degree %s not yet implemented"%K.degree() +