diff --git a/src/doc/de/tutorial/introduction.rst b/src/doc/de/tutorial/introduction.rst index 749b774d914..7b27528a897 100644 --- a/src/doc/de/tutorial/introduction.rst +++ b/src/doc/de/tutorial/introduction.rst @@ -45,7 +45,7 @@ Tutorial der richtige Ort um damit anzufangen. Zum Beispiel: over Rational Field sage: E.anlist(10) [0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3] - sage: E.rank() + sage: E.rank() # needs sage.libs.eclib 1 sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k diff --git a/src/doc/en/thematic_tutorials/explicit_methods_in_number_theory/birds_other.rst b/src/doc/en/thematic_tutorials/explicit_methods_in_number_theory/birds_other.rst index bdd5cc1d360..1575e93a2cb 100644 --- a/src/doc/en/thematic_tutorials/explicit_methods_in_number_theory/birds_other.rst +++ b/src/doc/en/thematic_tutorials/explicit_methods_in_number_theory/birds_other.rst @@ -123,6 +123,7 @@ more general, but doesn't scale nearly so well (yet). :: + sage: # needs sage.libs.eclib sage: M = CremonaModularSymbols(20014) # few seconds sage: M Cremona Modular Symbols space of dimension 5005 for diff --git a/src/doc/en/thematic_tutorials/explicit_methods_in_number_theory/elliptic_curves.rst b/src/doc/en/thematic_tutorials/explicit_methods_in_number_theory/elliptic_curves.rst index 6fe991abf45..5468125b22d 100644 --- a/src/doc/en/thematic_tutorials/explicit_methods_in_number_theory/elliptic_curves.rst +++ b/src/doc/en/thematic_tutorials/explicit_methods_in_number_theory/elliptic_curves.rst @@ -138,6 +138,7 @@ the :math:`5`-adic and :math:`997`-adic regulators of this curve. :: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('389a') sage: E.padic_regulator(5, 10) 5^2 + 2*5^3 + 2*5^4 + 4*5^5 + 3*5^6 + 4*5^7 + 3*5^8 + 5^9 + O(5^11) @@ -154,6 +155,7 @@ nontrivial computational challenge. Now in Sage computing the :: + sage: # needs sage.libs.eclib sage: E.padic_regulator(100003,5) # a couple of seconds 42582*100003^2 + 35250*100003^3 + 12790*100003^4 + 64078*100003^5 + O(100003^6) @@ -176,6 +178,7 @@ rank :math:`2`. :: + sage: # needs.sage.libs.eclib sage: E = EllipticCurve('389a') sage: L = E.padic_lseries(5) sage: L @@ -199,7 +202,7 @@ unpublished papers of Wuthrich and me. sage: E.sha().bound() # so only 2 could divide sha [2] sage: E = EllipticCurve('37a1') # so only 2 could divide sha - sage: E.sha().bound() + sage: E.sha().bound() # needs sage.libs.eclib ([2], 1) sage: E = EllipticCurve('389a1') sage: E.sha().bound() @@ -223,6 +226,7 @@ and :math:`7` do not divide the Shafarevich-Tate group of our rank :: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('389a1') sage: sha = E.sha() sage: sha.p_primary_bound(5) # iwasawa theory ==> 5 doesn't divide sha @@ -242,6 +246,7 @@ number in general! :: + sage: # needs sage.libs.eclib sage: E.sha().an() 1.00000000000000 @@ -257,7 +262,7 @@ GP scripts for computing Mordell-Weil groups of elliptic curves. sage: E = EllipticCurve([1,2,5,17,159]) sage: E.conductor() # not in the Tables 10272987 - sage: E.gens() # a few seconds + sage: E.gens() # a few seconds, needs sage.libs.eclib [(-3 : 9 : 1), (-3347/3249 : 1873597/185193 : 1)] @@ -274,6 +279,7 @@ the only free open source implementation available. :: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([1,2,5,7,17]) sage: E.integral_points(both_signs=True) [(1 : -9 : 1), (1 : 3 : 1)] @@ -283,6 +289,7 @@ rank :math:`3`, which has 36 integral points. :: + sage: # needs sage.libs.eclib sage: E = elliptic_curves.rank(3)[0] sage: E.integral_points(both_signs=True) # less than 3 seconds [(-3 : -1 : 1), (-3 : 0 : 1), (-2 : -4 : 1), (-2 : 3 : 1), ...(816 : -23310 : 1), (816 : 23309 : 1)] @@ -299,6 +306,7 @@ points). :: + sage: # needs sage.libs.eclib sage: E = elliptic_curves.rank(4)[0] sage: E.integral_points(both_signs=True) # about a minute [(-10 : 3 : 1), (-10 : 7 : 1), ... diff --git a/src/doc/en/tutorial/introduction.rst b/src/doc/en/tutorial/introduction.rst index 44eecb132c3..60b7b6b156d 100644 --- a/src/doc/en/tutorial/introduction.rst +++ b/src/doc/en/tutorial/introduction.rst @@ -42,7 +42,7 @@ tutorial is the place to start. For example: over Rational Field sage: E.anlist(10) [0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3] - sage: E.rank() + sage: E.rank() # needs sage.libs.eclib 1 sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k diff --git a/src/doc/es/tutorial/introduction.rst b/src/doc/es/tutorial/introduction.rst index 92e01281ef6..2be2afc682c 100644 --- a/src/doc/es/tutorial/introduction.rst +++ b/src/doc/es/tutorial/introduction.rst @@ -41,7 +41,7 @@ lugar justo para empezar. Por ejemplo: over Rational Field sage: E.anlist(10) [0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3] - sage: E.rank() + sage: E.rank() # needs sage.libs.eclib 1 sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k diff --git a/src/doc/fr/tutorial/introduction.rst b/src/doc/fr/tutorial/introduction.rst index cfc10a965a3..cb0bca5c794 100644 --- a/src/doc/fr/tutorial/introduction.rst +++ b/src/doc/fr/tutorial/introduction.rst @@ -44,7 +44,7 @@ tutoriel est le bon endroit où commencer. Voici quelques exemples : over Rational Field sage: E.anlist(10) [0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3] - sage: E.rank() + sage: E.rank() # needs sage.libs.eclib 1 sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k diff --git a/src/doc/it/tutorial/introduction.rst b/src/doc/it/tutorial/introduction.rst index 856aa26d484..e32eb78aac2 100644 --- a/src/doc/it/tutorial/introduction.rst +++ b/src/doc/it/tutorial/introduction.rst @@ -39,7 +39,7 @@ Per esempio: over Rational Field sage: E.anlist(10) [0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3] - sage: E.rank() + sage: E.rank() # needs sage.libs.eclib 1 sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k diff --git a/src/doc/ja/tutorial/introduction.rst b/src/doc/ja/tutorial/introduction.rst index 517e1690c6a..aa84a451b59 100644 --- a/src/doc/ja/tutorial/introduction.rst +++ b/src/doc/ja/tutorial/introduction.rst @@ -38,7 +38,7 @@ Sageのかなりの部分がPythonを使って実装されているものの， over Rational Field sage: E.anlist(10) [0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3] - sage: E.rank() + sage: E.rank() # needs.sage.libs.eclib 1 sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k diff --git a/src/doc/pt/tutorial/introduction.rst b/src/doc/pt/tutorial/introduction.rst index 8ddf257d3c7..fea32680b75 100644 --- a/src/doc/pt/tutorial/introduction.rst +++ b/src/doc/pt/tutorial/introduction.rst @@ -43,7 +43,7 @@ certo para começar. Por exemplo: over Rational Field sage: E.anlist(10) [0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3] - sage: E.rank() + sage: E.rank() # needs sage.libs.eclib 1 sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k diff --git a/src/doc/ru/tutorial/introduction.rst b/src/doc/ru/tutorial/introduction.rst index fe8681d8a79..2a9640e5e2a 100644 --- a/src/doc/ru/tutorial/introduction.rst +++ b/src/doc/ru/tutorial/introduction.rst @@ -44,7 +44,7 @@ over Rational Field sage: E.anlist(10) [0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3] - sage: E.rank() + sage: E.rank() # needs sage.libs.eclib 1 sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k diff --git a/src/doc/zh/tutorial/introduction.rst b/src/doc/zh/tutorial/introduction.rst index b694444dc58..899a6f79c5a 100644 --- a/src/doc/zh/tutorial/introduction.rst +++ b/src/doc/zh/tutorial/introduction.rst @@ -37,7 +37,7 @@ Python 初学者指南 [PyB]_ 列出了许多选择。如果你只是想快速 over Rational Field sage: E.anlist(10) [0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3] - sage: E.rank() + sage: E.rank() # needs sage.libs.eclib 1 sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k diff --git a/src/sage/features/mwrank.py b/src/sage/features/mwrank.py new file mode 100644 index 00000000000..996b6dfbc3c --- /dev/null +++ b/src/sage/features/mwrank.py @@ -0,0 +1,34 @@ +# sage_setup: distribution = sagemath-environment +r""" +Feature for testing the presence of the ``mwrank`` program (part +of eclib) +""" + +from . import Executable, FeatureTestResult + + +class Mwrank(Executable): + r""" + A :class:`~sage.features.Feature` describing the presence of + the ``mwrank`` program. + + EXAMPLES:: + + sage: from sage.features.mwrank import Mwrank + sage: Mwrank().is_present() # needs mwrank + FeatureTestResult('mwrank', True) + """ + def __init__(self): + r""" + TESTS:: + + sage: from sage.features.mwrank import Mwrank + sage: isinstance(Mwrank(), Mwrank) + True + """ + Executable.__init__(self, 'mwrank', executable='mwrank', + spkg='eclib', type='standard') + + +def all_features(): + return [Mwrank()] diff --git a/src/sage/features/sagemath.py b/src/sage/features/sagemath.py index 05179c410fd..e245acf57f4 100644 --- a/src/sage/features/sagemath.py +++ b/src/sage/features/sagemath.py @@ -537,6 +537,33 @@ def __init__(self): spkg='sagemath_ntl', type='standard') +class sage__libs__eclib(JoinFeature): + r""" + A :class:`sage.features.Feature` describing the presence of + :mod:`sage.libs.eclib`. + + In addition to the modularization purposes that this tag serves, + it also provides attribution to the upstream project. + + TESTS:: + + sage: from sage.features.sagemath import sage__libs__eclib + sage: sage__libs__eclib().is_present() # needs sage.libs.eclib + FeatureTestResult('sage.libs.eclib', True) + """ + def __init__(self): + r""" + TESTS:: + + sage: from sage.features.sagemath import sage__libs__eclib + sage: isinstance(sage__libs__eclib(), sage__libs__eclib) + True + """ + JoinFeature.__init__(self, 'sage.libs.eclib', + [PythonModule('sage.libs.eclib.mwrank')], + spkg='eclib', type='standard') + + class sage__libs__giac(JoinFeature): r""" A :class:`sage.features.Feature` describing the presence of :mod:`sage.libs.giac`. diff --git a/src/sage/interfaces/mwrank.py b/src/sage/interfaces/mwrank.py index 8d6af9b81d6..e28e7c6776c 100644 --- a/src/sage/interfaces/mwrank.py +++ b/src/sage/interfaces/mwrank.py @@ -1,3 +1,4 @@ +# sage.doctest: needs mwrank r""" Interface to mwrank """ diff --git a/src/sage/lfunctions/zero_sums.pyx b/src/sage/lfunctions/zero_sums.pyx index cefed0fd4ef..49e9269e1c4 100644 --- a/src/sage/lfunctions/zero_sums.pyx +++ b/src/sage/lfunctions/zero_sums.pyx @@ -1624,7 +1624,7 @@ cdef class LFunctionZeroSum_EllipticCurve(LFunctionZeroSum_abstract): 0 sage: E = EllipticCurve([-39,123]) - sage: E.rank() + sage: E.rank() # needs sage.libs.eclib 1 sage: Z = LFunctionZeroSum(E) sage: Z.analytic_rank_upper_bound(max_Delta=1) diff --git a/src/sage/libs/eclib/constructor.py b/src/sage/libs/eclib/constructor.py index 37df4eb935b..dacbbaf9c4c 100644 --- a/src/sage/libs/eclib/constructor.py +++ b/src/sage/libs/eclib/constructor.py @@ -1,3 +1,4 @@ +# sage.doctest: needs sage.libs.eclib "Cremona modular symbols" diff --git a/src/sage/libs/eclib/homspace.pyx b/src/sage/libs/eclib/homspace.pyx index f4baaee280a..d382ee2eefc 100644 --- a/src/sage/libs/eclib/homspace.pyx +++ b/src/sage/libs/eclib/homspace.pyx @@ -1,3 +1,4 @@ +# sage.doctest: needs sage.libs.eclib "Cremona modular symbols" from cysignals.signals cimport sig_on, sig_off diff --git a/src/sage/libs/eclib/interface.py b/src/sage/libs/eclib/interface.py index 35824051258..9dde679b229 100644 --- a/src/sage/libs/eclib/interface.py +++ b/src/sage/libs/eclib/interface.py @@ -1,3 +1,4 @@ +# sage.doctest: needs sage.libs.eclib r""" Sage interface to Cremona's ``eclib`` library (also known as ``mwrank``) diff --git a/src/sage/libs/eclib/mat.pyx b/src/sage/libs/eclib/mat.pyx index 989d9d1a70c..33c8ef2ac24 100644 --- a/src/sage/libs/eclib/mat.pyx +++ b/src/sage/libs/eclib/mat.pyx @@ -1,3 +1,4 @@ +# sage.doctest: needs sage.libs.eclib """ Cremona matrices """ diff --git a/src/sage/libs/eclib/mwrank.pyx b/src/sage/libs/eclib/mwrank.pyx index 4a2ca04e55f..34ddceed5d7 100644 --- a/src/sage/libs/eclib/mwrank.pyx +++ b/src/sage/libs/eclib/mwrank.pyx @@ -1,3 +1,4 @@ +# sage.doctest: needs sage.libs.eclib """ Cython interface to Cremona's ``eclib`` library (also known as ``mwrank``) diff --git a/src/sage/libs/eclib/newforms.pyx b/src/sage/libs/eclib/newforms.pyx index c067ef6acb9..2fb8ee5c52c 100644 --- a/src/sage/libs/eclib/newforms.pyx +++ b/src/sage/libs/eclib/newforms.pyx @@ -1,3 +1,4 @@ +# sage.doctest: needs sage.libs.eclib """ Modular symbols using eclib newforms """ diff --git a/src/sage/matrix/special.py b/src/sage/matrix/special.py index c864c4f07e0..9718033edcd 100644 --- a/src/sage/matrix/special.py +++ b/src/sage/matrix/special.py @@ -2012,7 +2012,7 @@ def block_matrix(*args, **kwds): ... ValueError: an element of parent was passed in, but only matrices, vectors and ring elements are accepted - sage: matrix.block([ + sage: matrix.block([ # needs sage.libs.eclib ....: [EllipticCurve('37a1').0], ....: ]) Traceback (most recent call last): diff --git a/src/sage/misc/benchmark.py b/src/sage/misc/benchmark.py index 4b9aa1e8b70..326c6a6b59f 100644 --- a/src/sage/misc/benchmark.py +++ b/src/sage/misc/benchmark.py @@ -22,6 +22,7 @@ def benchmark(n=-1): EXAMPLES:: + sage: # needs sage.libs.eclib sage: from sage.misc.benchmark import * sage: _ = benchmark() Running benchmark 0 @@ -105,6 +106,7 @@ def bench1(): BENCHMARK:: + sage: # needs sage.libs.eclib sage: from sage.misc.benchmark import * sage: print(bench1()[0]) Find the Mordell-Weil group of the elliptic curve 5077A using mwrank @@ -208,6 +210,7 @@ def bench7(): BENCHMARK:: + sage: # needs sage.libs.eclib sage: from sage.misc.benchmark import * sage: print(bench7()[0]) Compute the Mordell-Weil group of y^2 = x^3 + 37*x - 997. diff --git a/src/sage/misc/functional.py b/src/sage/misc/functional.py index 5c5e3c36e9b..d25da0741b8 100644 --- a/src/sage/misc/functional.py +++ b/src/sage/misc/functional.py @@ -1667,7 +1667,7 @@ def regulator(x): sage: x = polygen(ZZ, 'x') sage: regulator(NumberField(x^2 - 2, 'a')) # needs sage.rings.number_field 0.881373587019543 - sage: regulator(EllipticCurve('11a')) # needs sage.schemes + sage: regulator(EllipticCurve('11a')) # needs sage.schemes sage.libs.eclib 1.00000000000000 """ return x.regulator() diff --git a/src/sage/modular/modform/element.py b/src/sage/modular/modform/element.py index 4ba86a13e26..a30759c1252 100644 --- a/src/sage/modular/modform/element.py +++ b/src/sage/modular/modform/element.py @@ -926,6 +926,7 @@ def period(self, M, prec=53): These can be used to express the periods of `f` as exact linear combinations of the real and the imaginary period of `E`:: + sage: # needs sage.libs.eclib sage: s = E.modular_symbol(sign=+1) sage: t = E.modular_symbol(sign=-1, implementation='sage') sage: s(3/11), t(3/11) diff --git a/src/sage/modular/pollack_stevens/fund_domain.py b/src/sage/modular/pollack_stevens/fund_domain.py index 48f90638f03..38e9252efc1 100644 --- a/src/sage/modular/pollack_stevens/fund_domain.py +++ b/src/sage/modular/pollack_stevens/fund_domain.py @@ -1448,6 +1448,7 @@ def prep_hecke_on_gen(self, l, gen, modulus=None): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -1518,6 +1519,7 @@ def prep_hecke_on_gen_list(self, l, gen, modulus=None): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() diff --git a/src/sage/modular/pollack_stevens/manin_map.py b/src/sage/modular/pollack_stevens/manin_map.py index 84da85201e4..6ed5a07c1e6 100644 --- a/src/sage/modular/pollack_stevens/manin_map.py +++ b/src/sage/modular/pollack_stevens/manin_map.py @@ -10,6 +10,7 @@ EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi @@ -779,6 +780,7 @@ def hecke(self, ell, algorithm='prep'): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -831,6 +833,7 @@ def p_stabilize(self, p, alpha, V): EXAMPLES:: + sage: # needs.sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: f = phi._map diff --git a/src/sage/modular/pollack_stevens/modsym.py b/src/sage/modular/pollack_stevens/modsym.py index f708a44107d..52e6269e1aa 100644 --- a/src/sage/modular/pollack_stevens/modsym.py +++ b/src/sage/modular/pollack_stevens/modsym.py @@ -5,6 +5,7 @@ EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol(); phi Modular symbol of level 11 with values in Sym^0 Q^2 @@ -81,6 +82,7 @@ def _iterate_Up(Phi, p, M, ap, q, aq, check): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('57a') sage: p = 5 sage: prec = 4 @@ -121,6 +123,7 @@ def __init__(self, actor, MSspace): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: g = phi._map._codomain._act._Sigma0(matrix(ZZ,2,2,[1,2,3,4])) @@ -136,6 +139,7 @@ def _act_(self, g, sym): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: g = phi._map._codomain._act._Sigma0(matrix(ZZ,2,2,[2,1,5,-1])) @@ -153,6 +157,7 @@ def __init__(self, map_data, parent, construct=False): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a') sage: phi = E.pollack_stevens_modular_symbol() """ @@ -168,6 +173,7 @@ def _repr_(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi._repr_() @@ -183,6 +189,7 @@ def dict(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: Set([x.moment(0) for x in phi.dict().values()]) == Set([-1/5, 1, 0]) @@ -202,6 +209,7 @@ def weight(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.weight() @@ -217,6 +225,7 @@ def values(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -237,6 +246,7 @@ def _normalize(self, **kwds) -> Self: EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi._normalize() @@ -256,6 +266,7 @@ def _richcmp_(self, other, op): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi == phi @@ -280,6 +291,7 @@ def _add_(self, right): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -297,6 +309,7 @@ def _lmul_(self, right): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -314,6 +327,7 @@ def _rmul_(self, right): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -331,6 +345,7 @@ def _sub_(self, right): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -411,6 +426,7 @@ def plus_part(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -433,6 +449,7 @@ def minus_part(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -473,6 +490,7 @@ def hecke(self, ell, algorithm='prep'): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -485,7 +503,6 @@ def hecke(self, ell, algorithm='prep'): True sage: phi.hecke(101) == phi * E.ap(101) True - sage: all(phi.hecke(p, algorithm='naive') == phi * E.ap(p) for p in [2,3,5,101]) # long time True """ @@ -507,6 +524,7 @@ def valuation(self, p=None): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -524,6 +542,7 @@ def valuation(self, p=None): ... ValueError: you must specify a prime + sage: # needs sage.libs.eclib sage: phi2 = phi.lift(11, M=2) sage: phi2.valuation() 0 @@ -547,6 +566,7 @@ def diagonal_valuation(self, p): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -579,6 +599,7 @@ def is_Tq_eigensymbol(self, q, p=None, M=None): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -625,6 +646,7 @@ def Tq_eigenvalue(self, q, p=None, M=None, check=True): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.values() @@ -632,7 +654,6 @@ def Tq_eigenvalue(self, q, p=None, M=None, check=True): sage: phi_ord = phi.p_stabilize(p = 3, ap = E.ap(3), M = 10, ordinary = True) sage: phi_ord.Tq_eigenvalue(2,3,10) + 2 O(3^10) - sage: phi_ord.Tq_eigenvalue(3,3,10) 2 + 3^2 + 2*3^3 + 2*3^4 + 2*3^6 + 3^8 + 2*3^9 + O(3^10) sage: phi_ord.Tq_eigenvalue(3,3,100) @@ -692,6 +713,7 @@ def is_ordinary(self, p=None, P=None) -> bool: EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: phi = E.pollack_stevens_modular_symbol() sage: phi.is_ordinary(2) @@ -752,22 +774,25 @@ def evaluate_twisted(self, a, chi): EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('17a1') - sage: L = E.padic_lseries(5, implementation='pollackstevens', precision=4) #long time - sage: D = L.quadratic_twist() # long time - sage: L.symbol().evaluate_twisted(1,D) # long time + sage: L = E.padic_lseries(5, implementation='pollackstevens', precision=4) + sage: D = L.quadratic_twist() + sage: L.symbol().evaluate_twisted(1,D) (1 + 5 + 3*5^2 + 5^3 + O(5^4), 5^2 + O(5^3), 1 + O(5^2), 2 + O(5)) + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('40a4') - sage: L = E.padic_lseries(7, implementation='pollackstevens', precision=4) #long time - sage: D = L.quadratic_twist() # long time - sage: L.symbol().evaluate_twisted(1,D) # long time + sage: L = E.padic_lseries(7, implementation='pollackstevens', precision=4) + sage: D = L.quadratic_twist() + sage: L.symbol().evaluate_twisted(1,D) (4 + 6*7 + 3*7^2 + O(7^4), 6*7 + 6*7^2 + O(7^3), 6 + O(7^2), 1 + O(7)) TESTS: Check for :issue:`32878`:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: L = E.padic_lseries(3, implementation='pollackstevens', precision=4) sage: D = 5 @@ -798,6 +823,7 @@ def _consistency_check(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a1') sage: phi = E.pollack_stevens_modular_symbol() sage: phi._consistency_check() @@ -882,6 +908,7 @@ def _find_alpha(self, p, k, M=None, ap=None, new_base_ring=None, ordinary=True, EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: p = 5 sage: M = 10 @@ -992,6 +1019,7 @@ def p_stabilize(self, p=None, M=20, alpha=None, ap=None, new_base_ring=None, ord EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: p = 5 sage: prec = 4 @@ -1010,6 +1038,7 @@ def p_stabilize(self, p=None, M=20, alpha=None, ap=None, new_base_ring=None, ord sage: phis.Tq_eigenvalue(5,M = 3) 1 + 4*5 + 3*5^2 + O(5^3) + sage: # needs sage.libs.eclib sage: phis = phi.p_stabilize(p,M = prec,ordinary=False) sage: phis.Tq_eigenvalue(5) 5 + 5^2 + 2*5^3 + O(5^5) @@ -1149,6 +1178,7 @@ def lift(self, p=None, M=None, alpha=None, new_base_ring=None, EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: f = E.pollack_stevens_modular_symbol() sage: g = f.lift(11,4,algorithm='stevens',eigensymbol=True) @@ -1161,11 +1191,13 @@ def lift(self, p=None, M=None, alpha=None, new_base_ring=None, We check that lifting and then specializing gives back the original symbol:: + sage: # needs sage.libs.eclib sage: g.specialize() == f True Another example, which showed precision loss in an earlier version of the code:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a') sage: p = 5 sage: prec = 4 @@ -1176,6 +1208,7 @@ def lift(self, p=None, M=None, alpha=None, new_base_ring=None, Another example:: + sage: # needs sage.libs.eclib sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries sage: E = EllipticCurve('37a') sage: p = 5 @@ -1188,6 +1221,7 @@ def lift(self, p=None, M=None, alpha=None, new_base_ring=None, Examples using Greenberg's algorithm:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: phi = E.pollack_stevens_modular_symbol() sage: Phi = phi.lift(11,8,algorithm='greenberg',eigensymbol=True) @@ -1283,6 +1317,7 @@ def _lift_to_OMS(self, p, M, new_base_ring, algorithm='greenberg'): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: f = E.pollack_stevens_modular_symbol() sage: f._lift_to_OMS(11,4,Qp(11,4)) @@ -1362,6 +1397,7 @@ def _find_aq(self, p, M, check): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: f = E.pollack_stevens_modular_symbol() sage: f._find_aq(5,10,True) @@ -1408,6 +1444,7 @@ def _find_extraprec(self, p, M, alpha, check): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: p = 5 sage: M = 10 @@ -1473,14 +1510,15 @@ def p_stabilize_and_lift(self, p, M, alpha=None, ap=None, EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: f = E.pollack_stevens_modular_symbol() - sage: g = f.p_stabilize_and_lift(3,10) # long time - sage: g.Tq_eigenvalue(5) # long time + sage: g = f.p_stabilize_and_lift(3,10) + sage: g.Tq_eigenvalue(5) 1 + O(3^10) - sage: g.Tq_eigenvalue(7) # long time + sage: g.Tq_eigenvalue(7) 1 + 2*3 + 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + O(3^10) - sage: g.Tq_eigenvalue(3) # long time + sage: g.Tq_eigenvalue(3) 2 + 3^2 + 2*3^3 + 2*3^4 + 2*3^6 + 3^8 + 2*3^9 + O(3^10) """ if check: @@ -1577,11 +1615,12 @@ def padic_lseries(self, *args, **kwds): EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('37a') sage: phi = E.pollack_stevens_modular_symbol() - sage: L = phi.lift(37, M=6, eigensymbol=True).padic_lseries(); L # long time + sage: L = phi.lift(37, M=6, eigensymbol=True).padic_lseries(); L 37-adic L-series of Modular symbol of level 37 with values in Space of 37-adic distributions with k=0 action and precision cap 7 - sage: L.series(2) # long time + sage: L.series(2) O(37^6) + (4 + 37 + 36*37^2 + 19*37^3 + 21*37^4 + O(37^5))*T + O(T^2) """ from sage.modular.pollack_stevens.padic_lseries import pAdicLseries diff --git a/src/sage/modular/pollack_stevens/padic_lseries.py b/src/sage/modular/pollack_stevens/padic_lseries.py index defacb1b357..c17637c4a6c 100644 --- a/src/sage/modular/pollack_stevens/padic_lseries.py +++ b/src/sage/modular/pollack_stevens/padic_lseries.py @@ -46,23 +46,25 @@ class pAdicLseries(SageObject): EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('37a') sage: p = 5 sage: prec = 4 - sage: L = E.padic_lseries(p, implementation='pollackstevens', precision=prec) # long time - sage: L[1] # long time + sage: L = E.padic_lseries(p, implementation='pollackstevens', precision=prec) + sage: L[1] 1 + 4*5 + 2*5^2 + O(5^3) - sage: L.series(3) # long time + sage: L.series(3) O(5^4) + (1 + 4*5 + 2*5^2 + O(5^3))*T + (3 + O(5^2))*T^2 + O(T^3) :: + sage: # long time, needs sage.libs.eclib sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries sage: E = EllipticCurve('20a') sage: phi = E.pollack_stevens_modular_symbol() - sage: Phi = phi.p_stabilize_and_lift(3, 4) # long time - sage: L = pAdicLseries(Phi) # long time - sage: L.series(4) # long time + sage: Phi = phi.p_stabilize_and_lift(3, 4) + sage: L = pAdicLseries(Phi) + sage: L.series(4) 2*3 + O(3^4) + (3 + O(3^2))*T + (2 + O(3))*T^2 + O(3^0)*T^3 + O(T^4) An example of a `p`-adic `L`-series associated to a modular @@ -92,17 +94,17 @@ def __init__(self, symb, gamma=None, quadratic_twist=1, precision=None): EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries sage: E = EllipticCurve('11a3') sage: phi = E.pollack_stevens_modular_symbol() sage: p = 11 sage: prec = 3 - sage: Phi = phi.lift(p, prec,eigensymbol=True) # long time - sage: L = pAdicLseries(Phi) # long time - sage: L.series(3) # long time + sage: Phi = phi.lift(p, prec,eigensymbol=True) + sage: L = pAdicLseries(Phi) + sage: L.series(3) O(11^3) + (2 + 5*11 + O(11^2))*T + (10 + O(11))*T^2 + O(T^3) - - sage: TestSuite(L).run() # long time + sage: TestSuite(L).run() """ self._coefficients = {} @@ -129,11 +131,12 @@ def __getitem__(self, n): EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('14a5') - sage: L = E.padic_lseries(7,implementation='pollackstevens',precision=5) # long time - sage: L[0] # long time + sage: L = E.padic_lseries(7,implementation='pollackstevens',precision=5) + sage: L[0] O(7^5) - sage: L[1] # long time + sage: L[1] 5 + 5*7 + 2*7^2 + 2*7^3 + O(7^4) """ if n in self._coefficients: @@ -172,9 +175,10 @@ def __eq__(self, other): EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('11a') - sage: L = E.padic_lseries(11,implementation='pollackstevens',precision=6) # long time - sage: L == loads(dumps(L)) # indirect doctest, long time + sage: L = E.padic_lseries(11,implementation='pollackstevens',precision=6) + sage: L == loads(dumps(L)) # indirect doctest True """ if not isinstance(other, pAdicLseries): @@ -191,9 +195,10 @@ def __ne__(self, other): EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('11a') - sage: L = E.padic_lseries(11,implementation='pollackstevens',precision=6) # long time - sage: L != L # long time + sage: L = E.padic_lseries(11,implementation='pollackstevens',precision=6) + sage: L != L False """ return not self.__eq__(other) @@ -204,15 +209,16 @@ def symbol(self): EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries sage: E = EllipticCurve('21a4') sage: phi = E.pollack_stevens_modular_symbol() - sage: Phi = phi.p_stabilize_and_lift(2,5) # long time - sage: L = pAdicLseries(Phi) # long time - sage: L.symbol() # long time + sage: Phi = phi.p_stabilize_and_lift(2,5) + sage: L = pAdicLseries(Phi) + sage: L.symbol() Modular symbol of level 42 with values in Space of 2-adic distributions with k=0 action and precision cap 15 - sage: L.symbol() is Phi # long time + sage: L.symbol() is Phi True """ return self._symb @@ -223,9 +229,10 @@ def prime(self): EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('19a') - sage: L = E.padic_lseries(19, implementation='pollackstevens',precision=6) # long time - sage: L.prime() # long time + sage: L = E.padic_lseries(19, implementation='pollackstevens',precision=6) + sage: L.prime() 19 """ return self._symb.parent().prime() @@ -236,6 +243,7 @@ def quadratic_twist(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries sage: E = EllipticCurve('37a') sage: phi = E.pollack_stevens_modular_symbol() @@ -252,9 +260,10 @@ def _repr_(self): EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('14a2') - sage: L = E.padic_lseries(3, implementation='pollackstevens', precision=4) # long time - sage: L._repr_() # long time + sage: L = E.padic_lseries(3, implementation='pollackstevens', precision=4) + sage: L._repr_() '3-adic L-series of Modular symbol of level 42 with values in Space of 3-adic distributions with k=0 action and precision cap 8' """ @@ -272,16 +281,18 @@ def series(self, prec=5): EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('14a2') sage: p = 3 sage: prec = 6 - sage: L = E.padic_lseries(p,implementation='pollackstevens',precision=prec) # long time - sage: L.series(4) # long time + sage: L = E.padic_lseries(p,implementation='pollackstevens',precision=prec) + sage: L.series(4) 2*3 + 3^4 + 3^5 + O(3^6) + (2*3 + 3^2 + O(3^4))*T + (2*3 + O(3^2))*T^2 + (3 + O(3^2))*T^3 + O(T^4) + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve("15a3") - sage: L = E.padic_lseries(5,implementation='pollackstevens',precision=15) # long time - sage: L.series(3) # long time + sage: L = E.padic_lseries(5,implementation='pollackstevens',precision=15) + sage: L.series(3) O(5^15) + (2 + 4*5^2 + 3*5^3 + 5^5 + 2*5^6 + 3*5^7 + 3*5^8 + 2*5^9 + 2*5^10 + 3*5^11 + 5^12 + O(5^13))*T + (4*5 + 4*5^3 + 3*5^4 + 4*5^5 + 3*5^6 + 2*5^7 + 5^8 + 4*5^9 + 3*5^10 + O(5^11))*T^2 + O(T^3) sage: E = EllipticCurve("79a1") @@ -316,13 +327,16 @@ def interpolation_factor(self, ap, chip=1, psi=None): EXAMPLES:: sage: E = EllipticCurve('19a2') - sage: L = E.padic_lseries(3,implementation='pollackstevens',precision=6) # long time - sage: ap = E.ap(3) # long time - sage: L.interpolation_factor(ap) # long time + + sage: # long time, needs sage.libs.eclib + sage: L = E.padic_lseries(3,implementation='pollackstevens',precision=6) + sage: ap = E.ap(3) + sage: L.interpolation_factor(ap) 3^2 + 3^3 + 2*3^5 + 2*3^6 + O(3^7) Comparing against a different implementation:: + sage: # needs sage.libs.eclib sage: L = E.padic_lseries(3) sage: (1-1/L.alpha(prec=4))^2 3^2 + 3^3 + O(3^5) @@ -357,10 +371,11 @@ def _basic_integral(self, a, j): EXAMPLES:: + sage: # long time, needs sage.libs.eclib sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries sage: E = EllipticCurve('11a3') - sage: L = E.padic_lseries(5, implementation='pollackstevens', precision=4) #long time - sage: L._basic_integral(1,2) # long time + sage: L = E.padic_lseries(5, implementation='pollackstevens', precision=4) + sage: L._basic_integral(1,2) 2*5^2 + 5^3 + O(5^4) """ symb = self.symbol() diff --git a/src/sage/modular/pollack_stevens/space.py b/src/sage/modular/pollack_stevens/space.py index fb357341d2e..cd94f2b2162 100644 --- a/src/sage/modular/pollack_stevens/space.py +++ b/src/sage/modular/pollack_stevens/space.py @@ -48,6 +48,7 @@ :: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a1') sage: phi = E.pollack_stevens_modular_symbol(); phi Modular symbol of level 37 with values in Sym^0 Q^2 @@ -858,6 +859,7 @@ def ps_modsym_from_elliptic_curve(E, sign=0, implementation='eclib'): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('113a1') sage: symb = E.pollack_stevens_modular_symbol() # indirect doctest sage: symb @@ -865,6 +867,7 @@ def ps_modsym_from_elliptic_curve(E, sign=0, implementation='eclib'): sage: symb.values() [-1/2, 1, -1, 0, 0, 1, 1, -1, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, 0] + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0,1]) sage: symb = E.pollack_stevens_modular_symbol() sage: symb.values() @@ -975,7 +978,7 @@ def ps_modsym_from_simple_modsym_space(A, name='alpha'): sage: from sage.modular.pollack_stevens.space import ps_modsym_from_simple_modsym_space sage: E = EllipticCurve('11a') - sage: f_E = E.pollack_stevens_modular_symbol(); f_E.values() + sage: f_E = E.pollack_stevens_modular_symbol(); f_E.values() # needs sage.libs.eclib [-1/5, 1, 0] sage: A = ModularSymbols(11, sign=1, weight=2).decomposition()[0] sage: f_plus = ps_modsym_from_simple_modsym_space(A); f_plus.values() @@ -991,7 +994,7 @@ def ps_modsym_from_simple_modsym_space(A, name='alpha'): sage: (-1/5)*vector(QQ, f_plus.values()) + (1/2)*vector(QQ, f_minus.values()) (-1/5, 1, 0) - sage: vector(QQ, f_E.values()) + sage: vector(QQ, f_E.values()) # needs sage.libs.eclib (-1/5, 1, 0) The next few examples all illustrate the ways in which exceptions are diff --git a/src/sage/rings/padics/padic_template_element.pxi b/src/sage/rings/padics/padic_template_element.pxi index 9bc8db04203..c6a19b52bbb 100644 --- a/src/sage/rings/padics/padic_template_element.pxi +++ b/src/sage/rings/padics/padic_template_element.pxi @@ -537,7 +537,7 @@ cdef class pAdicTemplateElement(pAdicGenericElement): Check to see that :issue:`10292` is resolved:: - sage: # needs sage.schemes + sage: # needs sage.schemes sage.libs.eclib sage: E = EllipticCurve('37a') sage: R = E.padic_regulator(7) sage: len(R.expansion()) diff --git a/src/sage/schemes/elliptic_curves/BSD.py b/src/sage/schemes/elliptic_curves/BSD.py index 5ed20b13ed1..828519b636f 100644 --- a/src/sage/schemes/elliptic_curves/BSD.py +++ b/src/sage/schemes/elliptic_curves/BSD.py @@ -86,6 +86,7 @@ def mwrank_two_descent_work(E, two_tor_rk) -> tuple: EXAMPLES:: + sage: # needs sage.libs.eclib sage: from sage.schemes.elliptic_curves.BSD import mwrank_two_descent_work sage: E = EllipticCurve('14a') sage: mwrank_two_descent_work(E, E.two_torsion_rank()) @@ -130,7 +131,7 @@ def pari_two_descent_work(E) -> tuple: sage: pari_two_descent_work(E) (0, 0, 0, 0, []) sage: E = EllipticCurve('37a') - sage: pari_two_descent_work(E) # random, up to sign + sage: pari_two_descent_work(E) # random, up to sign, needs sage.libs.eclib (1, 1, 0, 0, [(0 : -1 : 1)]) sage: E = EllipticCurve('210e7') sage: pari_two_descent_work(E) @@ -215,6 +216,7 @@ def heegner_index_work(E) -> tuple: EXAMPLES:: + sage: # needs sage.libs.eclib sage: from sage.schemes.elliptic_curves.BSD import heegner_index_work sage: heegner_index_work(EllipticCurve('14a')) (1, -31) @@ -298,6 +300,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, EXAMPLES:: + sage: # needs sage.libs.eclib sage: EllipticCurve('11a').prove_BSD(verbosity=2) p = 2: True by 2-descent True for p not in {2, 5} by Kolyvagin. @@ -305,6 +308,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, True for p = 5 by Kolyvagin bound [] + sage: # needs sage.libs.eclib sage: EllipticCurve('14a').prove_BSD(verbosity=2) p = 2: True by 2-descent True for p not in {2, 3} by Kolyvagin. @@ -312,6 +316,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, True for p = 3 by Kolyvagin bound [] + sage: # needs sage.libs.eclib sage: E = EllipticCurve("20a1") sage: E.prove_BSD(verbosity=2) p = 2: True by 2-descent @@ -319,6 +324,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, Kato further implies that #Sha[3] is trivial. [] + sage: # needs sage.libs.eclib sage: E = EllipticCurve("50b1") sage: E.prove_BSD(verbosity=2) p = 2: True by 2-descent @@ -337,16 +343,19 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, We know nothing with proof=True:: + sage: # needs sage.libs.eclib sage: E.prove_BSD() Set of all prime numbers: 2, 3, 5, 7, ... We (think we) know everything with proof=False:: + sage: # needs sage.libs.eclib sage: E.prove_BSD(proof=False) [] A curve of rank 0 and prime conductor:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('19a') sage: E.prove_BSD(verbosity=2) p = 2: True by 2-descent @@ -355,6 +364,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, True for p = 3 by Kolyvagin bound [] + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a') sage: E.rank() 1 @@ -370,6 +380,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, We test the consistency check for the 2-part of Sha:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a') sage: S = E.sha(); S Tate-Shafarevich group for the Elliptic Curve defined by y^2 + y = x^3 - x @@ -384,6 +395,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, An example with a Tamagawa number at 5:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('123a1') sage: E.prove_BSD(verbosity=2) p = 2: True by 2-descent @@ -394,8 +406,9 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, A curve for which 3 divides the order of the Tate-Shafarevich group:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('681b') - sage: E.prove_BSD(verbosity=2) # long time + sage: E.prove_BSD(verbosity=2) p = 2: True by 2-descent... True for p not in {2, 3} by Kolyvagin.... Remaining primes: @@ -406,6 +419,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, A curve for which we need to use ``heegner_index_bound``:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('198b') sage: E.prove_BSD(verbosity=1, secs_hi=1) p = 2: True by 2-descent @@ -415,6 +429,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, The ``return_BSD`` option gives an object with detailed information about the proof:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('26b') sage: B = E.prove_BSD(return_BSD=True) sage: B.two_tor_rk @@ -432,6 +447,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, This was fixed by :issue:`8184` and :issue:`7575`:: + sage: # needs sage.libs.eclib sage: EllipticCurve('438e1').prove_BSD(verbosity=1) p = 2: True by 2-descent... True for p not in {2} by Kolyvagin. @@ -439,14 +455,16 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5, :: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('960d1') - sage: E.prove_BSD(verbosity=1) # long time (4s on sage.math, 2011) + sage: E.prove_BSD(verbosity=1) p = 2: True by 2-descent True for p not in {2} by Kolyvagin. [] :: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('66b3') sage: E.prove_BSD(two_desc="pari",verbosity=1) p = 2: True by 2-descent diff --git a/src/sage/schemes/elliptic_curves/constructor.py b/src/sage/schemes/elliptic_curves/constructor.py index 5a97d2eda9b..1777ba0116c 100644 --- a/src/sage/schemes/elliptic_curves/constructor.py +++ b/src/sage/schemes/elliptic_curves/constructor.py @@ -472,9 +472,11 @@ def create_object(self, version, key, *, names=None, **kwds): ``names`` is ignored at the moment, however it is used to support a convenient way to get a generator:: + sage: # needs sage.libs.eclib sage: E.

= EllipticCurve(QQ, [1, 3]) sage: P (-1 : 1 : 1) + sage: E.

= EllipticCurve(GF(5), [1, 3]) sage: P (4 : 1 : 1) @@ -958,13 +960,15 @@ def EllipticCurve_from_cubic(F, P=None, morphism=True): the generator:: sage: E = f.codomain() + sage: finv = f.inverse() + + sage: # needs sage.libs.eclib sage: E.label() '720e2' sage: E.rank() 1 sage: R = E.gens()[0]; R (-17280000/2527 : 9331200000/6859 : 1) - sage: finv = f.inverse() sage: [finv(k*R) for k in range(1,10)] [(-4 : 1 : 0), (-1 : 4 : 1), @@ -1469,6 +1473,7 @@ def EllipticCurves_with_good_reduction_outside_S(S=[], proof=None, verbose=False EXAMPLES:: + sage: # needs sage.libs.eclib sage: EllipticCurves_with_good_reduction_outside_S([]) [] sage: elist = EllipticCurves_with_good_reduction_outside_S([2]) @@ -1484,14 +1489,15 @@ def EllipticCurves_with_good_reduction_outside_S(S=[], proof=None, verbose=False Without ``Proof=False``, this example gives two warnings:: - sage: elist = EllipticCurves_with_good_reduction_outside_S([11], proof=False) # long time (14s on sage.math, 2011) - sage: len(elist) # long time + sage: # long time, needs sage.libs.eclib + sage: elist = EllipticCurves_with_good_reduction_outside_S([11], proof=False) + sage: len(elist) 12 - sage: ', '.join(e.label() for e in elist) # long time + sage: ', '.join(e.label() for e in elist) '11a1, 11a2, 11a3, 121a1, 121a2, 121b1, 121b2, 121c1, 121c2, 121d1, 121d2, 121d3' - sage: # long time - sage: elist = EllipticCurves_with_good_reduction_outside_S([2,3]) # long time (26s on sage.math, 2011) + sage: # long time, needs sage.libs.eclib + sage: elist = EllipticCurves_with_good_reduction_outside_S([2,3]) sage: len(elist) 752 sage: conds = sorted(set([e.conductor() for e in elist])) diff --git a/src/sage/schemes/elliptic_curves/ell_egros.py b/src/sage/schemes/elliptic_curves/ell_egros.py index 365070a3809..748a0a700c2 100644 --- a/src/sage/schemes/elliptic_curves/ell_egros.py +++ b/src/sage/schemes/elliptic_curves/ell_egros.py @@ -34,7 +34,8 @@ EXAMPLES: We find all elliptic curves with good reduction outside 2, listing the label of each:: - sage: [e.label() for e in EllipticCurves_with_good_reduction_outside_S([2])] # long time (5s on sage.math, 2013) + sage: # long time, needs sage.libs.eclib + sage: [e.label() for e in EllipticCurves_with_good_reduction_outside_S([2])] ['32a1', '32a2', '32a3', @@ -65,7 +66,8 @@ installed: two of the auxiliary curves whose Mordell-Weil bases are required have conductors 13068 and 52272 so are in the database):: - sage: [e.label() for e in EllipticCurves_with_good_reduction_outside_S([11], proof=False)] # long time (13s on sage.math, 2011) + sage: # long time, needs sage.libs.eclib + sage: [e.label() for e in EllipticCurves_with_good_reduction_outside_S([11], proof=False)] ['11a1', '11a2', '11a3', '121a1', '121a2', '121b1', '121b2', '121c1', '121c2', '121d1', '121d2', '121d3'] AUTHORS: @@ -319,6 +321,7 @@ def egros_from_jlist(jlist, S=[]): EXAMPLES:: + sage: # needs sage.libs.eclib sage: from sage.schemes.elliptic_curves.ell_egros import egros_get_j, egros_from_jlist sage: jlist=egros_get_j([3]) sage: elist=egros_from_jlist(jlist,[3]) @@ -371,14 +374,17 @@ def egros_get_j(S=[], proof=None, verbose=False): EXAMPLES:: + sage: # needs sage.libs.eclib sage: from sage.schemes.elliptic_curves.ell_egros import egros_get_j sage: egros_get_j([]) [1728] - sage: egros_get_j([2]) # long time (3s on sage.math, 2013) + + sage: # long time, needs sage.libs.eclib + sage: egros_get_j([2]) [128, 432, -864, 1728, 3375/2, -3456, 6912, 8000, 10976, -35937/4, 287496, -784446336, -189613868625/128] - sage: egros_get_j([3]) # long time (3s on sage.math, 2013) + sage: egros_get_j([3]) [0, -576, 1536, 1728, -5184, -13824, 21952/9, -41472, 140608/3, -12288000] - sage: jlist=egros_get_j([2,3]); len(jlist) # long time (30s) + sage: jlist=egros_get_j([2,3]); len(jlist) 83 """ if not all(p.is_prime() for p in S): diff --git a/src/sage/schemes/elliptic_curves/ell_generic.py b/src/sage/schemes/elliptic_curves/ell_generic.py index 5f560c566b9..8d06c22cc7e 100644 --- a/src/sage/schemes/elliptic_curves/ell_generic.py +++ b/src/sage/schemes/elliptic_curves/ell_generic.py @@ -1511,7 +1511,7 @@ def gens(self): ... NotImplementedError: not implemented. sage: E = EllipticCurve(QQ, [1,1]) - sage: E.gens() + sage: E.gens() # needs sage.libs.eclib [(0 : 1 : 1)] """ raise NotImplementedError("not implemented.") @@ -2188,12 +2188,14 @@ def _multiple_x_numerator(self, n, x=None): :: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("37a") sage: P = E.gens()[0] sage: x = P[0] :: + sage: # needs sage.libs.eclib sage: (35*P)[0] -804287518035141565236193151/1063198259901027900600665796 sage: E._multiple_x_numerator(35, x) @@ -2203,6 +2205,7 @@ def _multiple_x_numerator(self, n, x=None): :: + sage: # needs sage.libs.eclib sage: (36*P)[0] 54202648602164057575419038802/15402543997324146892198790401 sage: E._multiple_x_numerator(36, x) @@ -2310,6 +2313,7 @@ def _multiple_x_denominator(self, n, x=None): :: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("43a") sage: P = E.gens()[0] sage: x = P[0] diff --git a/src/sage/schemes/elliptic_curves/ell_modular_symbols.py b/src/sage/schemes/elliptic_curves/ell_modular_symbols.py index b6bdccff654..b508fd45a81 100644 --- a/src/sage/schemes/elliptic_curves/ell_modular_symbols.py +++ b/src/sage/schemes/elliptic_curves/ell_modular_symbols.py @@ -34,11 +34,14 @@ EXAMPLES:: sage: E = EllipticCurve("19a1") + + sage: # needs sage.libs.eclib sage: m = E.modular_symbol() sage: m(0) 1/3 sage: m(1/17) -2/3 + sage: m2 = E.modular_symbol(-1, implementation='sage') sage: m2(0) 0 @@ -170,9 +173,11 @@ def sign(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: m = EllipticCurve('11a1').modular_symbol() sage: m.sign() 1 + sage: m = EllipticCurve('11a1').modular_symbol(sign=-1, implementation='sage') sage: m.sign() -1 @@ -185,6 +190,7 @@ def elliptic_curve(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: m = EllipticCurve('11a1').modular_symbol() sage: m.elliptic_curve() Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field @@ -197,6 +203,7 @@ def base_ring(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: m = EllipticCurve('11a1').modular_symbol() sage: m.base_ring() Rational Field @@ -209,10 +216,12 @@ def _repr_(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: m = EllipticCurve('11a1').modular_symbol() sage: m Modular symbol with sign 1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field + sage: m = EllipticCurve('43a1').modular_symbol(sign=-1, implementation='sage') sage: m Modular symbol with sign -1 over Rational Field attached to @@ -245,6 +254,7 @@ def __init__(self, E, sign, nap=1000): EXAMPLES:: + sage: # needs sage.libs.eclib sage: from sage.schemes.elliptic_curves.ell_modular_symbols import ModularSymbolECLIB sage: E = EllipticCurve('11a1') sage: M = ModularSymbolECLIB(E,+1) @@ -260,6 +270,7 @@ def __init__(self, E, sign, nap=1000): This is a rank 1 case with vanishing positive twists:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('121b1') sage: M = ModularSymbolECLIB(E,+1) sage: M(0) @@ -267,16 +278,19 @@ def __init__(self, E, sign, nap=1000): sage: M(1/7) 1/2 + sage: # needs sage.libs.eclib sage: M = EllipticCurve('121d1').modular_symbol(implementation='eclib') sage: M(0) 2 + sage: # needs sage.libs.eclib sage: E = EllipticCurve('15a1') sage: [C.modular_symbol(implementation='eclib')(0) for C in E.isogeny_class()] [1/4, 1/8, 1/4, 1/2, 1/8, 1/16, 1/2, 1] Since :issue:`10256`, the interface for negative modular symbols in eclib is available:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: Mplus = E.modular_symbol(+1); Mplus Modular symbol with sign 1 over Rational Field attached to @@ -291,6 +305,7 @@ def __init__(self, E, sign, nap=1000): The scaling factor relative to eclib's normalization is 1/2 for curves of negative discriminant:: + sage: # needs sage.libs.eclib sage: [E.discriminant() for E in cremona_curves([14])] [-21952, 941192, -1835008, -28, 25088, 98] sage: [E.modular_symbol()._scaling for E in cremona_curves([14])] @@ -298,6 +313,7 @@ def __init__(self, E, sign, nap=1000): TESTS (for :issue:`10236`):: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: m = E.modular_symbol(implementation='eclib') sage: m(1/7) @@ -310,6 +326,7 @@ def __init__(self, E, sign, nap=1000): v20210310 the value of ``nap`` is increased automatically by ``eclib``:: + sage: # needs sage.libs.eclib sage: from sage.schemes.elliptic_curves.ell_modular_symbols import ModularSymbolECLIB sage: E = EllipticCurve('1590g1') sage: m = ModularSymbolECLIB(E, sign=+1, nap=300) @@ -320,6 +337,7 @@ def __init__(self, E, sign, nap=1000): effect. The correct values may verified by the numerical implementation:: + sage: # needs sage.libs.eclib sage: m = ModularSymbolECLIB(E, sign=+1, nap=400) sage: [m(a/5) for a in [1..4]] [13/2, -13/2, -13/2, 13/2] @@ -346,6 +364,7 @@ def _call_with_caching(self, r, base_at_infinity=True): EXAMPLES:: + sage: # needs sage.libs.eclib sage: m = EllipticCurve('11a1').modular_symbol(implementation='eclib') sage: m._call_with_caching(0) 1/5 @@ -365,6 +384,7 @@ def __call__(self, r, base_at_infinity=True): EXAMPLES:: + sage: # needs sage.libs.eclib sage: m = EllipticCurve('11a1').modular_symbol(implementation='eclib') sage: m(0) 1/5 @@ -485,6 +505,14 @@ def _find_scaling_L_ratio(self): EXAMPLES:: + sage: # needs sage.libs.eclib + sage: m = EllipticCurve('37a1').modular_symbol() + sage: m._scaling + 1 + sage: m = EllipticCurve('389a1').modular_symbol() + sage: m._scaling + 1 + sage: m = EllipticCurve('11a1').modular_symbol(implementation='sage') sage: m._scaling 1/5 @@ -497,12 +525,6 @@ def _find_scaling_L_ratio(self): sage: m = EllipticCurve('37a1').modular_symbol(implementation='sage') sage: m._scaling -1 - sage: m = EllipticCurve('37a1').modular_symbol() - sage: m._scaling - 1 - sage: m = EllipticCurve('389a1').modular_symbol() - sage: m._scaling - 1 sage: m = EllipticCurve('389a1').modular_symbol(implementation='sage') sage: m._scaling 1 @@ -519,8 +541,9 @@ def _find_scaling_L_ratio(self): TESTS:: + sage: # long time, needs sage.libs.eclib sage: rk0 = ['11a1', '11a2', '15a1', '27a1', '37b1'] - sage: for la in rk0: # long time (3s on sage.math, 2011) + sage: for la in rk0: ....: E = EllipticCurve(la) ....: me = E.modular_symbol(implementation='eclib') ....: ms = E.modular_symbol(implementation='sage') @@ -531,10 +554,11 @@ def _find_scaling_L_ratio(self): 1/3 1/3 1/3 2/3 2/3 2/3 + sage: # long time, needs sage.libs.eclib sage: rk1 = ['37a1','43a1','53a1', '91b1','91b2','91b3'] sage: [EllipticCurve(la).modular_symbol()(0) for la in rk1] # long time (1s on sage.math, 2011) [0, 0, 0, 0, 0, 0] - sage: for la in rk1: # long time (8s on sage.math, 2011) + sage: for la in rk1: ....: E = EllipticCurve(la) ....: m = E.modular_symbol() ....: lp = E.padic_lseries(5) diff --git a/src/sage/schemes/elliptic_curves/ell_number_field.py b/src/sage/schemes/elliptic_curves/ell_number_field.py index 509cfac364a..becc5c01c68 100644 --- a/src/sage/schemes/elliptic_curves/ell_number_field.py +++ b/src/sage/schemes/elliptic_curves/ell_number_field.py @@ -152,6 +152,7 @@ def base_extend(self, R): Check that non-torsion points are remembered when extending the base field (see :issue:`16034`):: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([1, 0, 1, -1751, -31352]) sage: K. = QuadraticField(5) sage: E.gens() @@ -329,13 +330,14 @@ def height_pairing_matrix(self, points=None, precision=None, normalised=True): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0, 0, 1, -1, 0]) sage: E.height_pairing_matrix() [0.0511114082399688] For rank 0 curves, the result is a valid 0x0 matrix:: - sage: EllipticCurve('11a').height_pairing_matrix() + sage: EllipticCurve('11a').height_pairing_matrix() # needs sage.libs.eclib [] sage: E = EllipticCurve('5077a1') sage: E.height_pairing_matrix([E.lift_x(x) for x in [-2,-7/4,1]], precision=100) @@ -343,6 +345,7 @@ def height_pairing_matrix(self, points=None, precision=None, normalised=True): [ -1.3095767070865761992624519454 2.7173593928122930896610589220 1.0998184305667292139777571432] [-0.63486715783715592064475542573 1.0998184305667292139777571432 0.66820516565192793503314205089] + sage: # needs sage.libs.eclib sage: E = EllipticCurve('389a1') sage: E = EllipticCurve('389a1') sage: P, Q = E.point([-1,1,1]), E.point([0,-1,1]) @@ -352,6 +355,7 @@ def height_pairing_matrix(self, points=None, precision=None, normalised=True): Over a number field:: + sage: # needs sage.libs.eclib sage: x = polygen(QQ) sage: K. = NumberField(x^2 + 47) sage: EK = E.base_extend(K) @@ -490,6 +494,7 @@ def regulator_of_points(self, points=[], precision=None, normalised=True): :: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('389a1') sage: P,Q = E.gens() sage: E.regulator_of_points([P,Q]) @@ -2329,7 +2334,7 @@ def gens(self, **kwds): sage: x = polygen(ZZ, 'x') sage: K. = NumberField(x^2 + 23, 'a') sage: E = EllipticCurve(K,[0,0,0,101,0]) - sage: E.gens() + sage: E.gens() # needs sage.libs.eclib ((23831509/8669448*a - 2867471/8669448 : 76507317707/18049790736*a - 424166479633/18049790736 : 1), (-2031032029/969232392*a + 58813561/969232392 : -15575984630401/21336681877488*a + 451041199309/21336681877488 : 1), (-186948623/4656964 : 549438861195/10049728312*a : 1)) @@ -2337,6 +2342,7 @@ def gens(self, **kwds): It can happen that no points are found if the height bounds used in the search are too small (see :issue:`10745`):: + sage: # needs sage.libs.eclib sage: K. = NumberField(x^4 + x^2 - 7) sage: E = EllipticCurve(K, [1, 0, 5*t^2 + 16, 0, 0]) sage: E.gens(lim1=1, lim3=1) @@ -2348,6 +2354,7 @@ def gens(self, **kwds): Check that the point found has infinite order, and that it is on the curve:: + sage: # needs sage.libs.eclib sage: P=gg[0]; P.order() # long time +Infinity sage: E.defining_polynomial()(*P) # long time @@ -2355,6 +2362,7 @@ def gens(self, **kwds): Here is a curve of rank 2:: + sage: # needs sage.libs.eclib sage: K. = NumberField(x^2 - 17) sage: E = EllipticCurve(K, [-4, 0]) sage: E.gens() @@ -2364,6 +2372,7 @@ def gens(self, **kwds): Test that points of finite order are not included (see :issue:`13593`):: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("17a3") sage: K. = NumberField(x^2 + 3) sage: EK = E.base_extend(K) @@ -3368,6 +3377,7 @@ def lll_reduce(self, points, height_matrix=None, precision=None): Some examples over `\QQ`:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0, 1, 1, -2, 42]) sage: Pi = E.gens(); Pi [(-4 : 1 : 1), (-3 : 5 : 1), (-11/4 : 43/8 : 1), (-2 : 6 : 1)] @@ -3417,6 +3427,7 @@ def lll_reduce(self, points, height_matrix=None, precision=None): An example to show the explicit use of the height pairing matrix:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0, 1, 1, -2, 42]) sage: Pi = E.gens() sage: H = E.height_pairing_matrix(Pi,3) @@ -4055,13 +4066,14 @@ def gens_quadratic(self, **kwds): sage: E = EllipticCurve([1,2,3,40,50]) sage: E.conductor() 2123582 - sage: E.gens() + sage: E.gens() # needs sage.libs.eclib [(5 : 17 : 1)] sage: K. = QuadraticField(-1) sage: EK = E.change_ring(K) - sage: EK.gens_quadratic() + sage: EK.gens_quadratic() # needs sage.libs.eclib [(5 : 17 : 1), (-13 : 48*i + 5 : 1)] + sage: # needs sage.libs.eclib sage: E.change_ring(QuadraticField(3, 'a')).gens_quadratic() [(5 : 17 : 1), (-1 : 2*a - 1 : 1), (11/4 : 33/4*a - 23/8 : 1)] @@ -4069,16 +4081,16 @@ def gens_quadratic(self, **kwds): sage: E = EllipticCurve([0,0,0,197,0]) sage: E.conductor() 2483776 - sage: E.gens() + sage: E.gens() # needs sage.libs.eclib [(47995604297578081/7389879786648100 : -25038161802544048018837479/635266655830129794121000 : 1)] sage: K. = QuadraticField(7) - sage: E.change_ring(K).gens_quadratic() + sage: E.change_ring(K).gens_quadratic() # needs sage.libs.eclib [(-1209642055/59583566*a + 1639995844/29791783 : -377240626321899/1720892553212*a + 138577803462855/245841793316 : 1), (1/28 : 393/392*a : 1), (-61*a + 162 : 1098*a - 2916 : 1)] sage: E = EllipticCurve([1, a]) - sage: E.gens_quadratic() + sage: E.gens_quadratic() # needs sage.libs.eclib Traceback (most recent call last): ... ValueError: gens_quadratic() requires the elliptic curve to be a base change from Q diff --git a/src/sage/schemes/elliptic_curves/ell_point.py b/src/sage/schemes/elliptic_curves/ell_point.py index 16d1cd71c76..73c65a74f2b 100644 --- a/src/sage/schemes/elliptic_curves/ell_point.py +++ b/src/sage/schemes/elliptic_curves/ell_point.py @@ -539,6 +539,7 @@ class EllipticCurvePoint_field(EllipticCurvePoint, Test pickling an elliptic curve that has known points on it:: + sage: # needs sage.libs.eclib sage: e = EllipticCurve([0, 0, 1, -1, 0]); g = e.gens(); loads(dumps(e)) == e True @@ -788,6 +789,8 @@ def __bool__(self) -> bool: (0 : 1 : 0) sage: P.is_zero() True + + sage: # needs sage.libs.eclib sage: P = E.gens()[0] sage: P.is_zero() False @@ -992,6 +995,7 @@ def _add_(self, other): Example to show that bug :issue:`4820` is fixed:: + sage: # needs sage.libs.eclib sage: [type(c) for c in 2*EllipticCurve('37a1').gen(0)] [<... 'sage.rings.rational.Rational'>, <... 'sage.rings.rational.Rational'>, @@ -1083,6 +1087,7 @@ def _neg_(self): Example to show that bug :issue:`4820` is fixed:: + sage: # needs sage.libs.eclib sage: [type(c) for c in -EllipticCurve('37a1').gen(0)] [<... 'sage.rings.rational.Rational'>, <... 'sage.rings.rational.Rational'>, @@ -2717,6 +2722,7 @@ class EllipticCurvePoint_number_field(EllipticCurvePoint_field): Test pickling an elliptic curve that has known points on it:: + sage: # needs sage.libs.eclib sage: e = EllipticCurve([0, 0, 1, -1, 0]); g = e.gens(); loads(dumps(e)) == e True """ @@ -3032,14 +3038,16 @@ def has_good_reduction(self, P=None) -> bool: EXAMPLES:: sage: E = EllipticCurve('990e1') - sage: P = E.gen(0); P - (15 : 51 : 1) sage: [E.has_good_reduction(p) for p in [2,3,5,7]] [False, False, False, True] - sage: [P.has_good_reduction(p) for p in [2,3,5,7]] - [True, False, True, True] sage: [E.tamagawa_exponent(p) for p in [2,3,5,7]] [2, 2, 1, 1] + + sage: # needs sage.libs.eclib + sage: P = E.gen(0); P + (15 : 51 : 1) + sage: [P.has_good_reduction(p) for p in [2,3,5,7]] + [True, False, True, True] sage: [(2*P).has_good_reduction(p) for p in [2,3,5,7]] [True, True, True, True] sage: P.has_good_reduction() @@ -3242,7 +3250,7 @@ def height(self, precision=None, normalised=True, algorithm='pari'): 0.0511114082399688 sage: P.order() +Infinity - sage: E.regulator() + sage: E.regulator() # needs sage.libs.eclib 0.0511114082399688... sage: def naive_height(P): @@ -4396,6 +4404,8 @@ def log(self, base): sage: E = EllipticCurve([0,1,1,a,a]) sage: E.cardinality() 762 + + sage: # needs sage.libs.eclib sage: P = E.gens()[0] sage: Q = 400*P sage: Q.log(P) diff --git a/src/sage/schemes/elliptic_curves/ell_rational_field.py b/src/sage/schemes/elliptic_curves/ell_rational_field.py index 3e2387105ba..f1fcc0b880d 100644 --- a/src/sage/schemes/elliptic_curves/ell_rational_field.py +++ b/src/sage/schemes/elliptic_curves/ell_rational_field.py @@ -169,6 +169,7 @@ def __init__(self, ainvs, **kwds): right curve (see :issue:`10999`: the following used not to work when the large database was installed):: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('389a1') sage: [P.curve() is E for P in E.gens()] [True, True] @@ -225,7 +226,7 @@ def _set_rank(self, r): sage: E.rank() # returns bogus cached value 99 sage: E._EllipticCurve_rational_field__rank = None # undo the damage - sage: E.rank() # the correct rank + sage: E.rank() # the correct rank, needs sage.libs.eclib 1 """ self.__rank = (Integer(r), True) @@ -329,6 +330,7 @@ def _set_gens(self, gens): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('5077a1') sage: E.rank() 3 @@ -443,6 +445,7 @@ def mwrank(self, options=''): EXAMPLES:: + sage: # needs mwrank sage: E = EllipticCurve('37a1') sage: E.mwrank() # random ... @@ -464,14 +467,14 @@ def mwrank(self, options=''): Run mwrank with ``'verbose'`` flag set to 0 but list generators if found:: - sage: print(E.mwrank('-v0 -l')) + sage: print(E.mwrank('-v0 -l')) # needs mwrank Curve [0,0,0,877,0] : 0 <= rank <= 1 Regulator = 1 Run mwrank again, this time with a higher bound for point searching on homogeneous spaces:: - sage: print(E.mwrank('-v0 -l -b11')) + sage: print(E.mwrank('-v0 -l -b11')) # needs mwrank Curve [0,0,0,877,0] : Rank = 1 Generator 1 is [29604565304828237474403861024284371796799791624792913256602210:-256256267988926809388776834045513089648669153204356603464786949:490078023219787588959802933995928925096061616470779979261000]; height 95.98037... Regulator = 95.98037... @@ -512,13 +515,13 @@ def conductor(self, algorithm='pari'): sage: E = EllipticCurve([1, -1, 1, -29372, -1932937]) sage: E.conductor(algorithm='pari') 3006 - sage: E.conductor(algorithm='mwrank') + sage: E.conductor(algorithm='mwrank') # needs sage.libs.eclib 3006 sage: E.conductor(algorithm='gp') 3006 sage: E.conductor(algorithm='generic') 3006 - sage: E.conductor(algorithm='all') + sage: E.conductor(algorithm='all') # needs sage.libs.eclib 3006 .. NOTE:: @@ -774,6 +777,7 @@ def mwrank_curve(self, verbose=False): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: EE = E.mwrank_curve() sage: EE @@ -841,6 +845,7 @@ def two_descent(self, verbose=True, EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a1') sage: E.two_descent(verbose=False) True @@ -1094,6 +1099,7 @@ def _modular_symbol_normalize(self, sign, normalize, implementation, nap): TESTS:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a1') sage: E.modular_symbol(implementation = 'eclib') is E.modular_symbol(implementation = 'eclib', normalize = 'L_ratio') True @@ -1190,6 +1196,7 @@ def modular_symbol(self, sign=+1, normalize=None, implementation='eclib', nap=0) EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a1') sage: M = E.modular_symbol(); M Modular symbol with sign 1 over Rational Field attached to @@ -1221,6 +1228,7 @@ def modular_symbol(self, sign=+1, normalize=None, implementation='eclib', nap=0) Different curves in an isogeny class have modular symbols which differ by a nonzero rational factor:: + sage: # needs sage.libs.eclib sage: E1 = EllipticCurve('11a1') sage: M1 = E1.modular_symbol() sage: M1(0) @@ -1244,7 +1252,7 @@ def modular_symbol(self, sign=+1, normalize=None, implementation='eclib', nap=0) normalized:: sage: E = EllipticCurve('11a2') - sage: E.modular_symbol(implementation = 'eclib')(0) + sage: E.modular_symbol(implementation = 'eclib')(0) # needs sage.libs.eclib 1 sage: E.modular_symbol(implementation = 'sage', normalize='L_ratio')(0) 1 @@ -1257,7 +1265,7 @@ def modular_symbol(self, sign=+1, normalize=None, implementation='eclib', nap=0) while the non-normalized symbol is incorrect:: sage: E = EllipticCurve('11a3') - sage: E.modular_symbol(implementation = 'eclib')(0) + sage: E.modular_symbol(implementation = 'eclib')(0) # needs sage.libs.eclib 1/25 sage: E.modular_symbol(implementation = 'sage', normalize='none')(0) 1 @@ -1268,6 +1276,7 @@ def modular_symbol(self, sign=+1, normalize=None, implementation='eclib', nap=0) Since :issue:`10256`, the interface for negative modular symbols in eclib is available:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: Mplus = E.modular_symbol(+1); Mplus Modular symbol with sign 1 over Rational Field attached to @@ -1288,8 +1297,10 @@ def modular_symbol(self, sign=+1, normalize=None, implementation='eclib', nap=0) See :issue:`31443`:: sage: E = EllipticCurve('1590g1') - sage: m = E.modular_symbol(nap=300) # long time - sage: [m(a/5) for a in [1..4]] # long time + + sage: # long time, needs sage.libs.eclib + sage: m = E.modular_symbol(nap=300) + sage: [m(a/5) for a in [1..4]] [13/2, -13/2, -13/2, 13/2] These values are correct, as verified by the numerical @@ -1343,6 +1354,7 @@ def modular_symbol_numerical(self, sign=1, prec=20): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('19a1') sage: f = E.modular_symbol_numerical(1) sage: g = E.modular_symbol(1) @@ -1386,6 +1398,7 @@ def pollack_stevens_modular_symbol(self, sign=0, implementation='eclib'): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('113a1') sage: symb = E.pollack_stevens_modular_symbol() sage: symb @@ -1393,6 +1406,7 @@ def pollack_stevens_modular_symbol(self, sign=0, implementation='eclib'): sage: symb.values() [-1/2, 1, -1, 0, 0, 1, 1, -1, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, 0] + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0,1]) sage: symb = E.pollack_stevens_modular_symbol(+1) sage: symb.values() @@ -1679,6 +1693,7 @@ def analytic_rank_upper_bound(self, :: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("11a") sage: E.rank() 0 @@ -1694,6 +1709,7 @@ def analytic_rank_upper_bound(self, :: + sage: # needs sage.libs.eclib sage: for r in range(9): ....: E = elliptic_curves.rank(r)[0] ....: print((r, E.analytic_rank_upper_bound(max_Delta=1, @@ -1858,14 +1874,14 @@ def simon_two_descent(self, verbose=0, lim1=5, lim3=50, limtriv=3, See https://github.com/sagemath/sage/issues/38461 for details. (0, 0, []) sage: E = EllipticCurve('37a1') - sage: E.simon_two_descent() + sage: E.simon_two_descent() # needs sage.libs.eclib (1, 1, [(0 : 0 : 1)]) sage: E = EllipticCurve('389a1') sage: E._known_points = [] # clear cached points - sage: E.simon_two_descent() + sage: E.simon_two_descent() # needs sage.libs.eclib (2, 2, [(-3/4 : 7/8 : 1), (5/4 : 5/8 : 1)]) sage: E = EllipticCurve('5077a1') - sage: E.simon_two_descent() + sage: E.simon_two_descent() # needs sage.libs.eclib (3, 3, [(1 : 0 : 1), (2 : 0 : 1), (0 : 2 : 1)]) In this example Simon's program does not find any points, though it @@ -1882,24 +1898,26 @@ def simon_two_descent(self, verbose=0, lim1=5, lim3=50, limtriv=3, :: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve([1, -1, 0, -79, 289]) sage: E.simon_two_descent() (4, 4, [(6 : -1 : 1), (4 : 3 : 1), (5 : -2 : 1), (8 : 7 : 1)]) sage: E = EllipticCurve([0, 0, 1, -79, 342]) - sage: E.simon_two_descent() # long time (9s on sage.math, 2011) + sage: E.simon_two_descent() (5, 5, [(5 : 8 : 1), (10 : 23 : 1), (3 : 11 : 1), (-3 : 23 : 1), (0 : 18 : 1)]) sage: E = EllipticCurve([1, 1, 0, -2582, 48720]) sage: r, s, G = E.simon_two_descent(); r,s (6, 6) sage: E = EllipticCurve([0, 0, 0, -10012, 346900]) - sage: r, s, G = E.simon_two_descent(); r,s # long time + sage: r, s, G = E.simon_two_descent(); r,s (7, 7) sage: E = EllipticCurve([0, 0, 1, -23737, 960366]) - sage: r, s, G = E.simon_two_descent(); r,s # long time + sage: r, s, G = E.simon_two_descent(); r,s (8, 8) Example from :issue:`10832`:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([1,0,0,-6664,86543]) sage: E.simon_two_descent() (2, 3, [(-1/4 : 2377/8 : 1), (323/4 : 1891/8 : 1)]) @@ -1912,16 +1930,18 @@ def simon_two_descent(self, verbose=0, lim1=5, lim3=50, limtriv=3, despite that the algorithm has not found any points of infinite order :: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([1, 1, 0, -23611790086, 1396491910863060]) sage: E.simon_two_descent() (1, 2, []) sage: E.rank() 1 - sage: E.gens() # uses mwrank + sage: E.gens() [(4311692542083/48594841 : -13035144436525227/338754636611 : 1)] Example for :issue:`5153`:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([3,0]) sage: E.simon_two_descent() (1, 2, [(1 : 2 : 1)]) @@ -2051,6 +2071,7 @@ def rank(self, use_database=True, verbose=False, EXAMPLES:: + sage: # needs sage.libs.eclib sage: EllipticCurve('11a').rank() 0 sage: EllipticCurve('37a').rank() @@ -2068,6 +2089,7 @@ def rank(self, use_database=True, verbose=False, Examples with denominators in defining equations:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0, 0, 0, 0, -675/4]) sage: E.rank() 0 @@ -2079,6 +2101,7 @@ def rank(self, use_database=True, verbose=False, A large example where mwrank doesn't determine the result with certainty, but pari does:: + sage: # needs sage.libs.eclib sage: EllipticCurve([1,0,0,0,37455]).rank(proof=False) 0 sage: EllipticCurve([1,0,0,0,37455]).rank(proof=True) @@ -2097,6 +2120,7 @@ def rank(self, use_database=True, verbose=False, An example to check if the points are saturated:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0,0, 1, -7, 6]) sage: E.gens(use_database=False, algorithm='pari') # random [(2 : 0 : 1), (-1 : 3 : 1), (11 : 35 : 1)] @@ -2106,6 +2130,7 @@ def rank(self, use_database=True, verbose=False, Since :issue:`23962`, the default is to use the Cremona database. We also check that the result is cached correctly:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([-517, -4528]) # 1888b1 sage: E.rank(use_database=False) Traceback (most recent call last): @@ -2312,6 +2337,7 @@ def gens(self, proof=None, **kwds): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('389a') sage: E.gens() # random output [(-1 : 1 : 1), (0 : 0 : 1)] @@ -2323,12 +2349,14 @@ def gens(self, proof=None, **kwds): A non-integral example:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([-3/8,-2/3]) sage: E.gens() # random (up to sign) [(10/9 : 29/54 : 1)] A non-minimal example:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('389a1') sage: E1 = E.change_weierstrass_model([1/20,0,0,0]); E1 Elliptic Curve defined by y^2 + 8000*y = x^3 + 400*x^2 - 320000*x @@ -2340,6 +2368,7 @@ def gens(self, proof=None, **kwds): TESTS:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('389a') sage: len(E.gens()) 2 @@ -2356,7 +2385,7 @@ def gens(self, proof=None, **kwds): Check that :issue:`38813` has been fixed:: - sage: # long time + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve([-127^2,0]) sage: l = E.gens(use_database=False, algorithm='pari', pari_effort=4); l # random [(611429153205013185025/9492121848205441 : 15118836457596902442737698070880/924793900700594415341761 : 1)] @@ -2408,21 +2437,24 @@ def _compute_gens(self, proof, EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([-3/8, -2/3]) sage: gens, proved = E._compute_gens(proof=False) sage: proved True + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve([-127^2,0]) - sage: E.gens(use_database=False, algorithm='pari', pari_effort=4) # long time, random + sage: E.gens(use_database=False, algorithm='pari', pari_effort=4) # random [(611429153205013185025/9492121848205441 : 15118836457596902442737698070880/924793900700594415341761 : 1)] TESTS:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve([-127^2,0]) sage: P = E.lift_x(611429153205013185025/9492121848205441) - sage: ge = set(E.gens(use_database=False, algorithm='pari',pari_effort=4)) # long time - sage: ge <= set([P+T for T in E.torsion_points()] # long time + sage: ge = set(E.gens(use_database=False, algorithm='pari',pari_effort=4)) + sage: ge <= set([P+T for T in E.torsion_points()] ....: + [-P+T for T in E.torsion_points()]) True """ @@ -2561,6 +2593,7 @@ def gens_certain(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a1') sage: E.gens() # random (up to sign) [(0 : -1 : 1)] @@ -2569,6 +2602,7 @@ def gens_certain(self): TESTS:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a1') sage: P = E([0,-1]) sage: set(E.gens()) <= set([P,-P]) @@ -2598,15 +2632,18 @@ def ngens(self, proof=None): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a1') sage: E.ngens() 1 + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0,0,0,877,0]) sage: E.ngens() 1 - sage: print(E.mwrank('-v0 -b12 -l')) + sage: E = EllipticCurve([0,0,0,877,0]) + sage: print(E.mwrank('-v0 -b12 -l')) # needs mwrank Curve [0,0,0,877,0] : Rank = 1 Generator 1 is [29604565304828237474403861024284371796799791624792913256602210:-256256267988926809388776834045513089648669153204356603464786949:490078023219787588959802933995928925096061616470779979261000]; height 95.98037... Regulator = 95.980... @@ -2630,6 +2667,7 @@ def regulator(self, proof=None, precision=53, **kwds): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0, 0, 1, -1, 0]) sage: E.regulator() 0.0511114082399688 @@ -2743,6 +2781,7 @@ def saturation(self, points, verbose=False, max_prime=-1, min_prime=2): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a1') sage: P=E(0,0) sage: Q=5*P; Q @@ -2756,6 +2795,7 @@ def saturation(self, points, verbose=False, max_prime=-1, min_prime=2): ``v20190909``, this example would loop forever at default precision. Since version ``v20210310`` it runs fine:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([1, 0, 1, -977842, -372252745]) sage: P = E([-192128125858676194585718821667542660822323528626273/336995568430319276695106602174283479617040716649, 70208213492933395764907328787228427430477177498927549075405076353624188436/195630373799784831667835900062564586429333568841391304129067339731164107, 1]) sage: P.height() @@ -2773,6 +2813,7 @@ def saturation(self, points, verbose=False, max_prime=-1, min_prime=2): See :issue:`10840`. This used to cause eclib to crash since the curve is non-minimal at 2:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0,0,0,-13711473216,0]) sage: P = E([-19992,16313472]) sage: Q = E([-24108,-17791704]) @@ -2783,6 +2824,7 @@ def saturation(self, points, verbose=False, max_prime=-1, min_prime=2): See :issue:`34029`. With eclib versions prior to 20220621 this failed to saturate:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0, 0, 0, -17607, -889490]) sage: Q = E([-82,54]) sage: E.saturation([2*Q], max_prime=10) @@ -2843,6 +2885,7 @@ def CPS_height_bound(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("11a") sage: E.CPS_height_bound() 2.8774743273580445 @@ -2911,9 +2954,9 @@ def silverman_height_bound(self, algorithm='default'): sage: E = EllipticCurve('37a1') sage: E.silverman_height_bound() 4.825400758180918 - sage: E.silverman_height_bound(algorithm='mwrank') + sage: E.silverman_height_bound(algorithm='mwrank') # needs sage.libs.eclib 4.825400758180918 - sage: E.CPS_height_bound() + sage: E.CPS_height_bound() # needs sage.libs.eclib 0.16397076103046915 """ if algorithm == 'default': @@ -2969,13 +3012,13 @@ def point_search(self, height_limit, verbose=False, rank_bound=None): EXAMPLES:: sage: E = EllipticCurve('389a1') - sage: E.point_search(1, verbose=False) + sage: E.point_search(1, verbose=False) # needs sage.libs.eclib [(-1 : 1 : 1), (0 : 0 : 1)] Increasing the height_limit takes longer, but finds no more points:: - sage: E.point_search(10, verbose=False) # long time + sage: E.point_search(10, verbose=False) # long time, needs sage.libs.eclib [(-1 : 1 : 1), (0 : 0 : 1)] In fact this curve has rank 2 so no more than 2 points will ever be @@ -2983,7 +3026,7 @@ def point_search(self, height_limit, verbose=False, rank_bound=None): :: - sage: E.saturation(_) + sage: E.saturation(_) # needs sage.libs.eclib ([(-1 : 1 : 1), (0 : 0 : 1)], 1, 0.152460177943144) What this shows is that if the rank is 2 then the points listed do @@ -2991,12 +3034,12 @@ def point_search(self, height_limit, verbose=False, rank_bound=None): :: - sage: E.rank() + sage: E.rank() # needs sage.libs.eclib 2 If we only need one independent generator:: - sage: E.point_search(5, verbose=False, rank_bound=1) + sage: E.point_search(5, verbose=False, rank_bound=1) # needs sage.libs.eclib [(-2 : 0 : 1)] """ # Convert logarithmic height to height @@ -3030,7 +3073,7 @@ def selmer_rank(self, algorithm='pari'): sage: E = EllipticCurve([1, 1, 1, 508, -2551]) sage: E.selmer_rank() 4 - sage: E.selmer_rank(algorithm='mwrank') + sage: E.selmer_rank(algorithm='mwrank') # needs sage.libs.eclib 4 The following is the curve 960d1, which has rank 0, but @@ -3039,7 +3082,7 @@ def selmer_rank(self, algorithm='pari'): sage: E = EllipticCurve([0, -1, 0, -900, -10098]) sage: E.selmer_rank() 3 - sage: E.selmer_rank(algorithm='mwrank') + sage: E.selmer_rank(algorithm='mwrank') # needs sage.libs.eclib 3 This curve has rank 1, and 4 elements in Sha[2]. @@ -3055,7 +3098,7 @@ def selmer_rank(self, algorithm='pari'): sage: E = EllipticCurve([1, 0, 0, -49280, -4214808]) sage: E.selmer_rank() 5 - sage: E.rank() + sage: E.rank() # needs sage.libs.eclib 0 """ try: @@ -3099,7 +3142,7 @@ def rank_bound(self, algorithm='pari'): sage: E = EllipticCurve([0, -1, 1, -929, -10595]) sage: E.rank_bound() 0 - sage: E.rank_bound(algorithm='mwrank') + sage: E.rank_bound(algorithm='mwrank') # needs sage.libs.eclib 2 In the following last example, both algorithm only determine a rank bound larger than the actual rank:: @@ -3107,7 +3150,7 @@ def rank_bound(self, algorithm='pari'): sage: E = EllipticCurve([1, 1, 1, -896670, -327184905]) sage: E.rank_bound() 2 - sage: E.rank_bound(algorithm='mwrank') + sage: E.rank_bound(algorithm='mwrank') # needs sage.libs.eclib 2 sage: E.rank(only_use_mwrank=False) # uses L-function 0 @@ -5667,7 +5710,7 @@ def sha(self): sage: S Tate-Shafarevich group for the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field - sage: S.bound_kolyvagin() + sage: S.bound_kolyvagin() # needs sage.libs.eclib ([2], 1) """ try: @@ -6002,6 +6045,7 @@ def integral_x_coords_in_interval(self, xmin, xmax): The bug reported on :issue:`22719` is now fixed:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("141d1") sage: E.integral_points() [(0 : -1 : 1), (2 : -2 : 1)] @@ -6050,7 +6094,7 @@ def integral_points(self, mw_base='auto', both_signs=False, verbose=False): sage: E = EllipticCurve([0,0,1,-7,6]) sage: P1 = E.point((2,0)); P2 = E.point((-1,3)); P3 = E.point((4,6)) - sage: a = E.integral_points([P1,P2,P3]); a + sage: a = E.integral_points([P1,P2,P3]); a # needs sage.libs.eclib [(-3 : -1 : 1), (-2 : -4 : 1), (-1 : -4 : 1), (0 : -3 : 1), (1 : -1 : 1), (2 : -1 : 1), (3 : -4 : 1), (4 : -7 : 1), (8 : -22 : 1), (11 : -36 : 1), (14 : -52 : 1), (21 : -96 : 1), @@ -6059,6 +6103,7 @@ def integral_points(self, mw_base='auto', both_signs=False, verbose=False): :: + sage: # needs sage.libs.eclib sage: a = E.integral_points([P1,P2,P3], verbose=True) Using mw_basis [(2 : 0 : 1), (3 : -4 : 1), (8 : -22 : 1)] e1,e2,e3: -3.0124303725933... 1.0658205476962... 1.94660982489710 @@ -6078,6 +6123,7 @@ def integral_points(self, mw_base='auto', both_signs=False, verbose=False): :: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0,0,1,-7,6]) sage: a = E.integral_points(both_signs=True); a [(-3 : -1 : 1), (-3 : 0 : 1), (-2 : -4 : 1), (-2 : 3 : 1), (-1 : -4 : 1), @@ -6091,6 +6137,7 @@ def integral_points(self, mw_base='auto', both_signs=False, verbose=False): An example with negative discriminant:: + sage: # needs sage.libs.eclib sage: EllipticCurve('900d1').integral_points() [(-11 : -27 : 1), (-4 : -34 : 1), (4 : -18 : 1), (16 : -54 : 1)] @@ -6107,21 +6154,25 @@ def integral_points(self, mw_base='auto', both_signs=False, verbose=False): The bug reported on :issue:`4525` is now fixed:: + sage: # needs sage.libs.eclib sage: EllipticCurve('91b1').integral_points() [(-1 : -4 : 1), (1 : -1 : 1), (3 : -5 : 1)] :: - sage: [len(e.integral_points(both_signs=False)) for e in cremona_curves([11..100])] # long time (15s on sage.math, 2011) + sage: # long time, needs sage.libs.eclib + sage: [len(e.integral_points(both_signs=False)) for e in cremona_curves([11..100])] [2, 0, 2, 3, 2, 1, 3, 0, 2, 4, 2, 4, 3, 0, 0, 1, 2, 1, 2, 0, 2, 1, 0, 1, 3, 3, 1, 1, 4, 2, 3, 2, 0, 0, 5, 3, 2, 2, 1, 1, 1, 0, 1, 3, 0, 1, 0, 1, 1, 3, 6, 1, 2, 2, 2, 0, 0, 2, 3, 1, 2, 2, 1, 1, 0, 3, 2, 1, 0, 1, 0, 1, 3, 3, 1, 1, 5, 1, 0, 1, 1, 0, 1, 2, 0, 2, 0, 1, 1, 3, 1, 2, 2, 4, 4, 2, 1, 0, 0, 5, 1, 0, 1, 2, 0, 2, 2, 0, 0, 0, 1, 0, 3, 1, 5, 1, 2, 4, 1, 0, 1, 0, 1, 0, 1, 0, 2, 2, 0, 0, 1, 0, 1, 1, 4, 1, 0, 1, 1, 0, 4, 2, 0, 1, 1, 2, 3, 1, 1, 1, 1, 6, 2, 1, 1, 0, 2, 0, 6, 2, 0, 4, 2, 2, 0, 0, 1, 2, 0, 2, 1, 0, 3, 1, 2, 1, 4, 6, 3, 2, 1, 0, 2, 2, 0, 0, 5, 4, 1, 0, 0, 1, 0, 2, 2, 0, 0, 2, 3, 1, 3, 1, 1, 0, 1, 0, 0, 1, 2, 2, 0, 2, 0, 0, 1, 2, 0, 0, 4, 1, 0, 1, 1, 0, 1, 2, 0, 1, 4, 3, 1, 2, 2, 1, 1, 1, 1, 6, 3, 3, 3, 3, 1, 1, 1, 1, 1, 0, 7, 3, 0, 1, 3, 2, 1, 0, 3, 2, 1, 0, 2, 2, 6, 0, 0, 6, 2, 2, 3, 3, 5, 5, 1, 0, 6, 1, 0, 3, 1, 1, 2, 3, 1, 2, 1, 1, 0, 1, 0, 1, 0, 5, 5, 2, 2, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1] The bug reported at :issue:`4897` is now fixed:: + sage: # needs sage.libs.eclib sage: [P[0] for P in EllipticCurve([0,0,0,-468,2592]).integral_points()] [-24, -18, -14, -6, -3, 4, 6, 18, 21, 24, 36, 46, 102, 168, 186, 381, 1476, 2034, 67246] See :issue:`22063`:: + sage: # needs sage.libs.eclib sage: for n in [67,71,74,91]: # long time ....: assert 4*n^6 + 4*n^2 in [P[0] for P in EllipticCurve([0,0,0,2,n^2]).integral_points()] @@ -6541,12 +6592,14 @@ def S_integral_points(self, S, mw_base='auto', both_signs=False, verbose=False, :: + sage: # needs sage.libs.eclib sage: a = E.S_integral_points([2,3]) sage: len(a) 43 An example with negative discriminant:: + sage: # needs sage.libs.eclib sage: EllipticCurve('900d1').S_integral_points([17], both_signs=True) [(-11 : -27 : 1), (-11 : 27 : 1), (-4 : -34 : 1), (-4 : 34 : 1), (4 : -18 : 1), (4 : 18 : 1), (2636/289 : -98786/4913 : 1), (2636/289 : 98786/4913 : 1), @@ -6554,17 +6607,20 @@ def S_integral_points(self, S, mw_base='auto', both_signs=False, verbose=False, Output checked with Magma (corrected in 3 cases):: - sage: [len(e.S_integral_points([2], both_signs=False)) for e in cremona_curves([11..100])] # long time (17s on sage.math, 2011) + sage: # long time, needs sage.libs.eclib + sage: [len(e.S_integral_points([2], both_signs=False)) for e in cremona_curves([11..100])] [2, 0, 2, 3, 3, 1, 3, 1, 3, 5, 3, 5, 4, 1, 1, 2, 2, 2, 3, 1, 2, 1, 0, 1, 3, 3, 1, 1, 5, 3, 4, 2, 1, 1, 5, 3, 2, 2, 1, 1, 1, 0, 1, 3, 0, 1, 0, 1, 1, 3, 7, 1, 3, 3, 3, 1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 3, 3, 1, 1, 1, 0, 1, 3, 3, 1, 1, 7, 1, 0, 1, 1, 0, 1, 2, 0, 3, 1, 2, 1, 3, 1, 2, 2, 4, 5, 3, 2, 1, 1, 6, 1, 0, 1, 3, 1, 3, 3, 1, 1, 1, 1, 1, 3, 1, 5, 1, 2, 4, 1, 1, 1, 1, 1, 0, 1, 0, 2, 2, 0, 0, 1, 0, 1, 1, 6, 1, 0, 1, 1, 0, 4, 3, 1, 2, 1, 2, 3, 1, 1, 1, 1, 8, 3, 1, 2, 1, 2, 0, 8, 2, 0, 6, 2, 3, 1, 1, 1, 3, 1, 3, 2, 1, 3, 1, 2, 1, 6, 9, 3, 3, 1, 1, 2, 3, 1, 1, 5, 5, 1, 1, 0, 1, 1, 2, 3, 1, 1, 2, 3, 1, 3, 1, 1, 1, 1, 0, 0, 1, 3, 3, 1, 3, 1, 1, 2, 2, 0, 0, 6, 1, 0, 1, 1, 1, 1, 3, 1, 2, 6, 3, 1, 2, 2, 1, 1, 1, 1, 7, 5, 4, 3, 3, 1, 1, 1, 1, 1, 1, 8, 5, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 2, 3, 6, 1, 1, 7, 3, 3, 4, 5, 9, 6, 1, 0, 7, 1, 1, 3, 1, 1, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 7, 8, 2, 3, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1] An example from [PZGH1999]_:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0,0,0,-172,505]) sage: E.rank(), len(E.S_integral_points([3,5,7])) # long time (5s on sage.math, 2011) (4, 72) This is curve "7690e1" which failed until :issue:`4805` was fixed:: + sage: # needs sage.libs.eclib sage: EllipticCurve([1,1,1,-301,-1821]).S_integral_points([13,2]) [(-13 : -4 : 1), (-9 : -12 : 1), (-7 : 2 : 1), (21 : -52 : 1), (23 : -76 : 1), (63 : -516 : 1), (71 : -620 : 1), (87 : -844 : 1), @@ -7101,6 +7157,7 @@ def cremona_curves(conductors): EXAMPLES:: + sage: # needs sage.libs.eclib sage: [(E.label(), E.rank()) for E in cremona_curves(srange(35,40))] [('35a1', 0), ('35a2', 0), @@ -7135,6 +7192,7 @@ def cremona_optimal_curves(conductors): EXAMPLES:: + sage: # needs sage.libs.eclib sage: [(E.label(), E.rank()) for E in cremona_optimal_curves(srange(35,40))] [('35a1', 0), ('36a1', 0), @@ -7178,11 +7236,12 @@ def integral_points_with_bounded_mw_coeffs(E, mw_base, N, x_bound): We check that some large integral points in a paper of Zagier are found:: + sage: # long time, needs sage.libs.eclib sage: def t(a, b, x): # indirect doctest ....: E = EllipticCurve([0,0,0,a,b]) ....: xs = [P[0] for P in E.integral_points()] ....: return x in xs - sage: all(t(a,b,x) for a,b,x in [(-2,5, 1318), (4,-1, 4321), # long time + sage: all(t(a,b,x) for a,b,x in [(-2,5, 1318), (4,-1, 4321), ....: (0,17, 5234), (11,4, 16833), (-13,37, 60721), (-12,-10, 80327), ....: (-7,22, 484961), (-9,28, 764396), (-13,4, 1056517), (-19,-51, ....: 2955980), (-24,124, 4435710), (-30,133, 5143326), (-37,60, diff --git a/src/sage/schemes/elliptic_curves/ell_tate_curve.py b/src/sage/schemes/elliptic_curves/ell_tate_curve.py index 7defaf1d50c..11f22ec5f73 100644 --- a/src/sage/schemes/elliptic_curves/ell_tate_curve.py +++ b/src/sage/schemes/elliptic_curves/ell_tate_curve.py @@ -584,6 +584,7 @@ def padic_height(self, prec=20): EXAMPLES:: + sage: # needs sage.libs.eclib sage: e = EllipticCurve('130a1') sage: eq = e.tate_curve(5) sage: h = eq.padic_height(prec=10) @@ -593,6 +594,7 @@ def padic_height(self, prec=20): Check that it is a quadratic function:: + sage: # needs sage.libs.eclib sage: h(3*P)-3^2*h(P) O(5^9) """ @@ -641,6 +643,7 @@ def padic_regulator(self, prec=20): EXAMPLES:: + sage: # needs sage.libs.eclib sage: eq = EllipticCurve('130a1').tate_curve(5) sage: eq.padic_regulator() 2*5^-1 + 1 + 2*5 + 2*5^2 + 3*5^3 + 3*5^6 + 5^7 + 3*5^9 + 3*5^10 + 3*5^12 + 4*5^13 + 3*5^15 + 2*5^16 + 3*5^18 + 4*5^19 + 4*5^20 + 3*5^21 + 4*5^22 + O(5^23) diff --git a/src/sage/schemes/elliptic_curves/heegner.py b/src/sage/schemes/elliptic_curves/heegner.py index a4a8c26c71e..293471f60a1 100644 --- a/src/sage/schemes/elliptic_curves/heegner.py +++ b/src/sage/schemes/elliptic_curves/heegner.py @@ -69,7 +69,7 @@ sage: P = E.heegner_point(-7) sage: z = P.point_exact(); z == E(0, 1, 1) or -z == E(0, 1, 1) True - sage: E.regulator() + sage: E.regulator() # needs sage.libs.eclib 0.0498083972980648 sage: z.height() 0.448275575682583 @@ -82,6 +82,7 @@ The above is consistent with the following analytic computation:: + sage: # needs sage.libs.eclib sage: E.heegner_index(-7) 3.0000? """ @@ -3231,9 +3232,9 @@ def _trace_index(self, *args, **kwds): sage: P = E.heegner_point(-19); y = P._trace_numerical_conductor_1() sage: [c.real() for c in y] [-1.2...e-16, -1.00000000000000, 1.00000000000000] - sage: -2*E.gens()[0] + sage: -2*E.gens()[0] # needs sage.libs.eclib (0 : -1 : 1) - sage: P._trace_index() + sage: P._trace_index() # needs sage.libs.eclib 2 sage: P = E.heegner_point(-68); P @@ -3241,7 +3242,7 @@ def _trace_index(self, *args, **kwds): sage: N(P) (0.219223593595584 - 1.87443160153148*I : -1.34232921921325 - 1.52356748877889*I : 1.00000000000000) - sage: P._trace_index() + sage: P._trace_index() # needs sage.libs.eclib 0 """ if self.conductor() != 1: @@ -3784,7 +3785,7 @@ def _trace_numerical_conductor_1(self, prec=53): 0.150298370947295 sage: P.height() 0.0375745927368238 - sage: E.heegner_index(-8) + sage: E.heegner_index(-8) # needs sage.libs.eclib 2.0000? sage: E.torsion_order() 1 @@ -4137,6 +4138,7 @@ def index(self, *args, **kwds): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67); P.index() 6 """ @@ -4187,6 +4189,7 @@ def point_exact(self, prec=53): A rank 1 curve:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67) sage: P.point_exact() (6 : -15 : 1) @@ -4199,6 +4202,7 @@ def point_exact(self, prec=53): A rank 0 curve:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1'); P = E.kolyvagin_point(-7) sage: P.point_exact() (-1/2*sqrt_minus_7 + 1/2 : -2*sqrt_minus_7 - 2 : 1) @@ -4334,6 +4338,8 @@ def _trace_exact_conductor_1(self, prec=53): sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67) sage: P.trace_to_real_numerical() (1.61355529131986 : -2.18446840788880 : 1.00000000000000) + + sage: # needs sage.libs.eclib sage: P._trace_exact_conductor_1() # the actual point we're reducing (1357/841 : -53277/24389 : 1) sage: (P._trace_exact_conductor_1().height() / E.regulator()).sqrt() @@ -4360,7 +4366,7 @@ def _recognize_point_over_QQ(self, P, n): sage: E = EllipticCurve('43a'); P = E.heegner_point(-20).kolyvagin_point() sage: PP = P.numerical_approx(); PP (0.000000000000000 : -1.00000000000000 : 1.00000000000000) - sage: P._recognize_point_over_QQ(PP, 4) + sage: P._recognize_point_over_QQ(PP, 4) # needs sage.libs.eclib (0 : -1 : 1) """ # Here is where we *should* implement the "method of Cremona @@ -4394,16 +4400,18 @@ def mod(self, p, prec=53): A Kolyvagin point on a rank 1 curve:: sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67) - sage: P.mod(2) + sage: P.mod(2) # needs sage.libs.eclib (1 : 1 : 1) - sage: P.mod(3) + sage: P.mod(3) # needs sage.libs.eclib (1 : 0 : 1) - sage: P.mod(5) + sage: P.mod(5) # needs sage.libs.eclib (2 : 2 : 1) - sage: P.mod(7) + sage: P.mod(7) # needs sage.libs.eclib (6 : 0 : 1) sage: P.trace_to_real_numerical() (1.61355529131986 : -2.18446840788880 : 1.00000000000000) + + sage: # needs sage.libs.eclib sage: P._trace_exact_conductor_1() # the actual point we're reducing (1357/841 : -53277/24389 : 1) sage: (P._trace_exact_conductor_1().height() / E.regulator()).sqrt() @@ -4413,9 +4421,9 @@ def mod(self, p, prec=53): rank 1), and we reduce it modulo several primes.:: sage: E = EllipticCurve('11a1'); P = E.kolyvagin_point(-7) - sage: P.mod(3,70) # long time (4s on sage.math, 2013) + sage: P.mod(3,70) # long time, needs sage.libs.eclib (1 : 2 : 1) - sage: P.mod(5,70) + sage: P.mod(5,70) # needs sage.libs.eclib (1 : 4 : 1) sage: P.mod(7,70) Traceback (most recent call last): @@ -4425,7 +4433,7 @@ def mod(self, p, prec=53): Traceback (most recent call last): ... ValueError: p must be coprime to conductors and discriminant - sage: P.mod(13,70) + sage: P.mod(13,70) # needs sage.libs.eclib (3 : 4 : 1) """ # check preconditions @@ -5678,11 +5686,13 @@ def kolyvagin_reduction_data(E, q, first_only=True): A rank 1 example:: + sage: # needs sage.libs.eclib sage: kolyvagin_reduction_data(EllipticCurve('37a1'), 3) (17, -7, 1, 52) A rank 3 example:: + sage: # needs sage.libs.eclib sage: kolyvagin_reduction_data(EllipticCurve('5077a1'), 3) (11, -47, 5, 4234) sage: H = heegner_points(5077, -47) @@ -5698,7 +5708,7 @@ def kolyvagin_reduction_data(E, q, first_only=True): present):: sage: E = elliptic_curves.rank(4)[0] - sage: kolyvagin_reduction_data(E,3) # long time + sage: kolyvagin_reduction_data(E,3) # long time, needs sage.libs.eclib (11, -71, 7, 293060) sage: H = heegner_points(293060, -71) sage: H.kolyvagin_conductors(1,4,E,3) @@ -5706,6 +5716,7 @@ def kolyvagin_reduction_data(E, q, first_only=True): The first rank 2 example:: + sage: # needs sage.libs.eclib sage: kolyvagin_reduction_data(EllipticCurve('389a'), 3) (5, -7, 1, 130) sage: kolyvagin_reduction_data(EllipticCurve('389a'), 3, first_only=False) @@ -5713,11 +5724,13 @@ def kolyvagin_reduction_data(E, q, first_only=True): A large `q = 7`:: + sage: # needs sage.libs.eclib sage: kolyvagin_reduction_data(EllipticCurve('1143c1'), 7, first_only=False) (13, 83, -59, 3, 1536, 10496) Additive reduction:: + sage: # needs sage.libs.eclib sage: kolyvagin_reduction_data(EllipticCurve('2350g1'), 5, first_only=False) (19, 239, -311, 19, 6480, 85680) """ @@ -6456,10 +6469,10 @@ def kolyvagin_point(self, D, c=ZZ(1), check=True): Kolyvagin point of discriminant -67 on elliptic curve of conductor 37 sage: P.numerical_approx() # abs tol 1e-14 (6.00000000000000 : -15.0000000000000 : 1.00000000000000) - sage: P.index() + sage: P.index() # needs sage.libs.eclib 6 sage: g = E((0,-1,1)) # a generator - sage: E.regulator() == E.regulator_of_points([g]) + sage: E.regulator() == E.regulator_of_points([g]) # needs sage.libs.eclib True sage: 6*g (6 : -15 : 1) @@ -6648,7 +6661,7 @@ def heegner_index(self, D, min_p=2, prec=5, descent_second_limit=12, sage: E = EllipticCurve('11a') sage: E.heegner_discriminants(50) [-7, -8, -19, -24, -35, -39, -40, -43] - sage: E.heegner_index(-7) + sage: E.heegner_index(-7) # needs sage.libs.eclib 1.00000? :: @@ -6656,13 +6669,14 @@ def heegner_index(self, D, min_p=2, prec=5, descent_second_limit=12, sage: E = EllipticCurve('37b') sage: E.heegner_discriminants(100) [-3, -4, -7, -11, -40, -47, -67, -71, -83, -84, -95] - sage: E.heegner_index(-95) # long time (1 second) + sage: E.heegner_index(-95) # long time, needs sage.libs.eclib 2.00000? This tests doing direct computation of the Mordell-Weil group. :: + sage: # needs sage.libs.eclib sage: EllipticCurve('675b').heegner_index(-11) 3.0000? @@ -6675,8 +6689,9 @@ def heegner_index(self, D, min_p=2, prec=5, descent_second_limit=12, The curve 681b returns the true index, which is `3`:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('681b') - sage: I = E.heegner_index(-8); I + sage: I = E.heegner_index(-8); 3.0000? In fact, whenever the returned index has a denominator of @@ -6689,6 +6704,7 @@ def heegner_index(self, D, min_p=2, prec=5, descent_second_limit=12, which can be used to fine tune the 2-descent used to compute the regulator of the twist:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([1,-1,0,-1228,-16267]) sage: E.heegner_index(-8) Traceback (most recent call last): @@ -6697,7 +6713,8 @@ def heegner_index(self, D, min_p=2, prec=5, descent_second_limit=12, However when we search higher, we find the points we need:: - sage: E.heegner_index(-8, descent_second_limit=16, check_rank=False) # long time + sage: # long time, needs sage.libs.eclib + sage: E.heegner_index(-8, descent_second_limit=16, check_rank=False) 2.00000? Two higher rank examples (of ranks 2 and 3):: @@ -6708,9 +6725,9 @@ def heegner_index(self, D, min_p=2, prec=5, descent_second_limit=12, sage: E = EllipticCurve('5077a') sage: E.heegner_index(-7) +Infinity - sage: E.heegner_index(-7, check_rank=False) + sage: E.heegner_index(-7, check_rank=False) # needs sage.libs.eclib 0.001? - sage: E.heegner_index(-7, check_rank=False).lower() == 0 + sage: E.heegner_index(-7, check_rank=False).lower() == 0 # needs sage.libs.eclib True """ if not self.satisfies_heegner_hypothesis(D): @@ -6860,6 +6877,7 @@ def heegner_index_bound(self, D=0, prec=5, max_height=None): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: E.heegner_index_bound() ([2], -7, 2) @@ -6964,6 +6982,8 @@ def _heegner_index_in_EK(self, D): We compute the index for a rank 2 curve and found that it is 2:: sage: E = EllipticCurve('389a') + + sage: # needs sage.libs.eclib sage: E._heegner_index_in_EK(-7) 2 @@ -7076,48 +7096,54 @@ def heegner_sha_an(self, D, prec=53): An example where E has conductor 11:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('11a') - sage: E.heegner_sha_an(-7) # long time + sage: E.heegner_sha_an(-7) 1.00000000000000 The cache works:: - sage: E.heegner_sha_an(-7) is E.heegner_sha_an(-7) # long time + sage: # long time, needs sage.libs.eclib + sage: E.heegner_sha_an(-7) is E.heegner_sha_an(-7) True Lower precision:: - sage: E.heegner_sha_an(-7,10) # long time + sage: # long time, needs sage.libs.eclib + sage: E.heegner_sha_an(-7,10) 1.0 Checking that the cache works for any precision:: - sage: E.heegner_sha_an(-7,10) is E.heegner_sha_an(-7,10) # long time + sage: # long time, needs sage.libs.eclib + sage: E.heegner_sha_an(-7,10) is E.heegner_sha_an(-7,10) True Next we consider a rank 1 curve with nontrivial Sha over the quadratic imaginary field `K`; however, there is no Sha for `E` over `\QQ` or for the quadratic twist of `E`:: + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('37a') - sage: E.heegner_sha_an(-40) # long time + sage: E.heegner_sha_an(-40) 4.00000000000000 - sage: E.quadratic_twist(-40).sha().an() # long time + sage: E.quadratic_twist(-40).sha().an() 1 - sage: E.sha().an() # long time + sage: E.sha().an() 1 A rank 2 curve:: - sage: E = EllipticCurve('389a') # long time - sage: E.heegner_sha_an(-7) # long time + sage: # long time, needs sage.libs.eclib + sage: E = EllipticCurve('389a') + sage: E.heegner_sha_an(-7) 1.00000000000000 If we remove the hypothesis that `E(K)` has rank 1 in Conjecture 2.3 in [GZ1986]_ page 311, then that conjecture is false, as the following example shows:: - sage: # long time + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve('65a') sage: E.heegner_sha_an(-56) 1.00000000000000 diff --git a/src/sage/schemes/elliptic_curves/height.py b/src/sage/schemes/elliptic_curves/height.py index b1efcc44904..e210ff1410b 100644 --- a/src/sage/schemes/elliptic_curves/height.py +++ b/src/sage/schemes/elliptic_curves/height.py @@ -2070,7 +2070,7 @@ def min(self, tol, n_max, verbose=False): sage: h = E.height_function() sage: h.min(.01, 5) 0.03987318057488725 - sage: E.gen(0).height() + sage: E.gen(0).height() # needs sage.libs.eclib 0.0511114082399688 After base change the lower bound can decrease:: @@ -2083,7 +2083,7 @@ def min(self, tol, n_max, verbose=False): sage: h = E.height_function() sage: h.min(0.1, 5) 0.05731275270029196 - sage: [P.height() for P in E.gens()] + sage: [P.height() for P in E.gens()] # needs sage.libs.eclib [0.686667083305587, 0.327000773651605] """ # The lcm of the exponents of all the component groups at diff --git a/src/sage/schemes/elliptic_curves/lseries_ell.py b/src/sage/schemes/elliptic_curves/lseries_ell.py index 74f5233c0e1..83619b1f4c8 100644 --- a/src/sage/schemes/elliptic_curves/lseries_ell.py +++ b/src/sage/schemes/elliptic_curves/lseries_ell.py @@ -843,7 +843,9 @@ def L_ratio(self): sage: EllipticCurve([0,0,0,-193^2,0]).sha().an() 4 - sage: EllipticCurve([1, 0, 1, -131, 558]).sha().an() # long time + + sage: # long time, needs sage.libs.eclib + sage: EllipticCurve([1, 0, 1, -131, 558]).sha().an() 1.00000000000000 ALGORITHM: Compute the root number. If it is -1 then `L(E,s)` diff --git a/src/sage/schemes/elliptic_curves/padic_lseries.py b/src/sage/schemes/elliptic_curves/padic_lseries.py index 6cb54115933..21631d983d5 100644 --- a/src/sage/schemes/elliptic_curves/padic_lseries.py +++ b/src/sage/schemes/elliptic_curves/padic_lseries.py @@ -99,6 +99,7 @@ class pAdicLseries(SageObject): An ordinary example:: + sage: # needs sage.libs.eclib sage: e = EllipticCurve('389a') sage: L = e.padic_lseries(5) sage: L.series(0) @@ -116,6 +117,7 @@ class pAdicLseries(SageObject): A prime p such that E[p] is reducible:: + sage: # needs sage.libs.eclib sage: L = EllipticCurve('11a').padic_lseries(5) sage: L.series(1) 5 + O(5^2) + O(T) @@ -126,6 +128,7 @@ class pAdicLseries(SageObject): An example showing the calculation of nontrivial Teichmueller twists:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: lp = E.padic_lseries(7) sage: lp.series(4,eta=1) @@ -137,6 +140,7 @@ class pAdicLseries(SageObject): (Note that the last series vanishes at `T = 0`, which is consistent with :: + sage: # needs sage.libs.eclib sage: E.quadratic_twist(-7).rank() 1 @@ -146,6 +150,7 @@ class pAdicLseries(SageObject): The load-dumps test:: + sage: # needs sage.libs.eclib sage: lp = EllipticCurve('11a').padic_lseries(5) sage: lp == loads(dumps(lp)) True @@ -166,6 +171,7 @@ def __init__(self, E, p, implementation='eclib', normalize='L_ratio'): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: Lp = E.padic_lseries(3) sage: Lp.series(2,prec=3) @@ -201,6 +207,7 @@ def __add_negative_space(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: lp = E.padic_lseries(5) sage: lp.modular_symbol(1/7,sign=-1) #indirect doctest @@ -214,6 +221,7 @@ def __richcmp__(self, other, op): TESTS:: + sage: # needs sage.libs.eclib sage: lp1 = EllipticCurve('11a1').padic_lseries(5) sage: lp2 = EllipticCurve('11a1').padic_lseries(7) sage: lp3 = EllipticCurve('11a2').padic_lseries(5) @@ -234,6 +242,7 @@ def elliptic_curve(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: L = EllipticCurve('11a').padic_lseries(5) sage: L.elliptic_curve() Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field @@ -246,6 +255,7 @@ def prime(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: L = EllipticCurve('11a').padic_lseries(5) sage: L.prime() 5 @@ -258,6 +268,7 @@ def _repr_(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: e = EllipticCurve('37a') sage: e.padic_lseries(3)._repr_() '3-adic L-series of Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field' @@ -297,6 +308,7 @@ def modular_symbol(self, r, sign=+1, quadratic_twist=+1): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: lp = E.padic_lseries(5) sage: [lp.modular_symbol(r) for r in [0,1/5,oo,1/11]] @@ -306,6 +318,7 @@ def modular_symbol(self, r, sign=+1, quadratic_twist=+1): sage: [lp.modular_symbol(r,quadratic_twist=-20) for r in [0,1/5,oo,1/11]] [1, 1, 0, 1/2] + sage: # needs sage.libs.eclib sage: E = EllipticCurve('20a1') sage: Et = E.quadratic_twist(-4) sage: lpt = Et.padic_lseries(5) @@ -387,6 +400,7 @@ def measure(self, a, n, prec, quadratic_twist=+1, sign=+1): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a') sage: L = E.padic_lseries(5) sage: L.measure(1,2, prec=9) @@ -396,6 +410,7 @@ def measure(self, a, n, prec, quadratic_twist=+1, sign=+1): sage: L.measure(1,2, quadratic_twist=-4,prec=15) 4 + 4*5 + 4*5^2 + 3*5^3 + 2*5^4 + 5^5 + 3*5^6 + 5^8 + 2*5^9 + 3*5^12 + 2*5^13 + 4*5^14 + O(5^15) + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: a = E.quadratic_twist(-3).padic_lseries(5).measure(1,2,prec=15) sage: b = E.padic_lseries(5).measure(1,2, quadratic_twist=-3,prec=15) @@ -465,6 +480,7 @@ def alpha(self, prec=20): An ordinary prime:: + sage: # needs sage.libs.eclib sage: L = E.padic_lseries(5) sage: alpha = L.alpha(10); alpha 3 + 2*5 + 4*5^2 + 2*5^3 + 5^4 + 4*5^5 + 2*5^7 + 5^8 + 5^9 + O(5^10) @@ -473,6 +489,7 @@ def alpha(self, prec=20): A supersingular prime:: + sage: # needs sage.libs.eclib sage: L = E.padic_lseries(3) sage: alpha = L.alpha(10); alpha alpha + O(alpha^21) @@ -481,6 +498,7 @@ def alpha(self, prec=20): A reducible prime:: + sage: # needs sage.libs.eclib sage: L = EllipticCurve('11a').padic_lseries(5) sage: L.alpha(5) 1 + 4*5 + 3*5^2 + 2*5^3 + 4*5^4 + O(5^5) @@ -534,6 +552,7 @@ def order_of_vanishing(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: L = EllipticCurve('11a').padic_lseries(3) sage: L.order_of_vanishing() 0 @@ -593,6 +612,7 @@ def teichmuller(self, prec): EXAMPLES:: + sage: # needs sage.libs.eclib sage: L = EllipticCurve('11a').padic_lseries(7) sage: L.teichmuller(1) [0, 1, 2, 3, 4, 5, 6] @@ -612,6 +632,7 @@ def _e_bounds(self, n, prec): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: Lp = E.padic_lseries(2) sage: Lp._e_bounds(1,10) @@ -641,6 +662,7 @@ def _get_series_from_cache(self, n, prec, D, eta): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: Lp = E.padic_lseries(5) sage: Lp._pAdicLseries__series = {} # clear cached series @@ -668,6 +690,7 @@ def _set_series_in_cache(self, n, prec, D, eta, f): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: Lp = E.padic_lseries(5) sage: Lp.series(3,prec=5) @@ -693,6 +716,7 @@ def _quotient_of_periods_to_twist(self, D): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37b1') sage: lp = E.padic_lseries(3) sage: lp._quotient_of_periods_to_twist(-20) @@ -710,6 +734,7 @@ def _quotient_of_periods_to_twist(self, D): sage: lp._quotient_of_periods_to_twist(12) 1 + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: Et = E.quadratic_twist(-3) sage: lpt = Et.padic_lseries(5) @@ -770,6 +795,7 @@ def series(self, n=2, quadratic_twist=+1, prec=5, eta=0): We compute some `p`-adic `L`-functions associated to the elliptic curve 11a:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: p = 3 sage: E.is_ordinary(p) @@ -782,6 +808,7 @@ def series(self, n=2, quadratic_twist=+1, prec=5, eta=0): `p`-adic `L`-function has an extra 0 (compared to the non `p`-adic `L`-function):: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: p = 11 sage: E.is_ordinary(p) @@ -792,6 +819,7 @@ def series(self, n=2, quadratic_twist=+1, prec=5, eta=0): We compute a `p`-adic `L`-function that vanishes to order 2:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('389a') sage: p = 3 sage: E.is_ordinary(p) @@ -806,6 +834,7 @@ def series(self, n=2, quadratic_twist=+1, prec=5, eta=0): Checks if the precision can be changed (:issue:`5846`):: + sage: # needs sage.libs.eclib sage: L.series(3,prec=4) O(3^5) + O(3^2)*T + (2 + 2*3 + O(3^2))*T^2 + (2 + O(3))*T^3 + O(T^4) sage: L.series(3,prec=6) @@ -814,6 +843,7 @@ def series(self, n=2, quadratic_twist=+1, prec=5, eta=0): Rather than computing the `p`-adic `L`-function for the curve '15523a1', one can compute it as a quadratic_twist:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('43a1') sage: lp = E.padic_lseries(3) sage: lp.series(2,quadratic_twist=-19) @@ -834,11 +864,13 @@ def series(self, n=2, quadratic_twist=+1, prec=5, eta=0): It should now also work with `p=2` (:issue:`20798`):: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("53a1") sage: lp = E.padic_lseries(2) sage: lp.series(7) O(2^8) + (1 + 2^2 + 2^3 + O(2^5))*T + (1 + 2^3 + O(2^4))*T^2 + (2^2 + 2^3 + O(2^4))*T^3 + (2 + 2^2 + O(2^3))*T^4 + O(T^5) + sage: # needs sage.libs.eclib sage: E = EllipticCurve("109a1") sage: lp = E.padic_lseries(2) sage: lp.series(6) @@ -987,6 +1019,7 @@ def is_ordinary(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: L = EllipticCurve('11a').padic_lseries(5) sage: L.is_ordinary() True @@ -1000,6 +1033,7 @@ def is_supersingular(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: L = EllipticCurve('11a').padic_lseries(5) sage: L.is_supersingular() False @@ -1024,6 +1058,7 @@ def _c_bound(self, sign=+1): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: Lp = E.padic_lseries(5) sage: Lp._c_bound() @@ -1032,16 +1067,17 @@ def _c_bound(self, sign=+1): 0 sage: EllipticCurve('11a3').padic_lseries(5)._c_bound() 2 - sage: EllipticCurve('11a3').padic_lseries(5, implementation='sage')._c_bound() - 2 sage: EllipticCurve('50b1').padic_lseries(3)._c_bound() 0 - sage: EllipticCurve('50b1').padic_lseries(3, implementation='sage')._c_bound() - 1 sage: l = EllipticCurve("11a1").padic_lseries(5) sage: ls = l.series(1,eta=1); sage: l._c_bound(sign=-1) 0 + + sage: EllipticCurve('11a3').padic_lseries(5, implementation='sage')._c_bound() + 2 + sage: EllipticCurve('50b1').padic_lseries(3, implementation='sage')._c_bound() + 1 """ E = self._E p = self._p @@ -1122,6 +1158,7 @@ def _prec_bounds(self, n, prec, sign=+1): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: Lp = E.padic_lseries(5) sage: Lp._prec_bounds(3,10) @@ -1133,6 +1170,7 @@ def _prec_bounds(self, n, prec, sign=+1): sage: Lp._prec_bounds(15,10) [+Infinity, 13, 13, 13, 13, 12, 12, 12, 12, 12] + sage: # needs sage.libs.eclib sage: Lp = E.padic_lseries(3) sage: Lp._prec_bounds(15,10) [+Infinity, 14, 14, 13, 13, 13, 13, 13, 13, 12] @@ -1184,6 +1222,7 @@ def series(self, n=3, quadratic_twist=+1, prec=5, eta=0): A supersingular example, where we must compute to higher precision to see anything:: + sage: # needs sage.libs.eclib sage: e = EllipticCurve('37a') sage: L = e.padic_lseries(3); L 3-adic L-series of Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field @@ -1196,6 +1235,7 @@ def series(self, n=3, quadratic_twist=+1, prec=5, eta=0): An example where we only compute the leading term (:issue:`15737`):: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("17a1") sage: L = E.padic_lseries(3) sage: L.series(4,prec=1) @@ -1203,6 +1243,7 @@ def series(self, n=3, quadratic_twist=+1, prec=5, eta=0): It works also for `p=2`:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("11a1") sage: lp = E.padic_lseries(2) sage: lp.series(10) @@ -1327,6 +1368,7 @@ def is_ordinary(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: L = EllipticCurve('11a').padic_lseries(19) sage: L.is_ordinary() False @@ -1340,6 +1382,7 @@ def is_supersingular(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: L = EllipticCurve('11a').padic_lseries(19) sage: L.is_supersingular() True @@ -1355,6 +1398,7 @@ def _prec_bounds(self, n, prec): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: Lp = E.padic_lseries(19) sage: Lp._prec_bounds(3,5) @@ -1378,6 +1422,7 @@ def _poly(self, a): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("14a1") sage: lp = E.padic_lseries(5) sage: K = lp.alpha().parent() @@ -1424,6 +1469,7 @@ def Dp_valued_series(self, n=3, quadratic_twist=+1, prec=5): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('14a') sage: L = E.padic_lseries(5) sage: L.Dp_valued_series(4) # long time (9s on sage.math, 2011) @@ -1469,6 +1515,7 @@ def frobenius(self, prec=20, algorithm='mw'): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('14a') sage: L = E.padic_lseries(5) sage: phi = L.frobenius(5) @@ -1531,12 +1578,14 @@ def __phi_bpr(self, prec=0): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: lp = E.padic_lseries(19) sage: lp.frobenius(prec=1,algorithm='approx') #indirect doctest [ O(19^0) 4*19^-1 + O(19^0)] [ 14 + O(19) O(19^0)] + sage: # needs sage.libs.eclib sage: E = EllipticCurve('17a1') sage: lp = E.padic_lseries(3) sage: lp.frobenius(prec=3,algorithm='approx') @@ -1618,6 +1667,7 @@ def bernardi_sigma_function(self, prec=20): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('14a') sage: L = E.padic_lseries(5) sage: L.bernardi_sigma_function(prec=5) # Todo: some sort of consistency check!? @@ -1657,6 +1707,7 @@ def Dp_valued_height(self, prec=20): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('53a') sage: L = E.padic_lseries(5) sage: h = L.Dp_valued_height(7) @@ -1713,6 +1764,7 @@ def Dp_valued_regulator(self, prec=20, v1=0, v2=0): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('43a') sage: L = E.padic_lseries(7) sage: L.Dp_valued_regulator(7) diff --git a/src/sage/schemes/elliptic_curves/padics.py b/src/sage/schemes/elliptic_curves/padics.py index 87f985e7052..b4e78033f92 100644 --- a/src/sage/schemes/elliptic_curves/padics.py +++ b/src/sage/schemes/elliptic_curves/padics.py @@ -131,6 +131,7 @@ def padic_lseries(self, p, normalize=None, implementation='eclib', EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a') sage: L = E.padic_lseries(5); L 5-adic L-series of Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field @@ -141,6 +142,7 @@ def padic_lseries(self, p, normalize=None, implementation='eclib', rank `0` and in each case verify the interpolation property for their leading coefficient (i.e., value at 0):: + sage: # needs sage.libs.eclib sage: e = EllipticCurve('11a') sage: ms = e.modular_symbol() sage: [ms(1/11), ms(1/3), ms(0), ms(oo)] @@ -166,6 +168,7 @@ def padic_lseries(self, p, normalize=None, implementation='eclib', Next consider the curve 37b:: + sage: # needs sage.libs.eclib sage: e = EllipticCurve('37b') sage: L = e.padic_lseries(3) sage: P = L.series(5) @@ -195,6 +198,7 @@ def padic_lseries(self, p, normalize=None, implementation='eclib', Finally, we can use the overconvergent method of Pollack-Stevens.:: + sage: # needs sage.libs.eclib sage: e = EllipticCurve('11a') sage: L = e.padic_lseries(3, implementation = 'pollackstevens', precision = 6) sage: L.series(5) @@ -204,6 +208,7 @@ def padic_lseries(self, p, normalize=None, implementation='eclib', Another example with a semistable prime.:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("11a1") sage: L = E.padic_lseries(11, implementation = 'pollackstevens', precision=3) sage: L[1] @@ -271,23 +276,27 @@ def padic_regulator(self, p, prec=20, height=None, check_hypotheses=True): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("37a") sage: E.padic_regulator(5, 10) 5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + O(5^10) An anomalous case:: + sage: # needs sage.libs.eclib sage: E.padic_regulator(53, 10) 26*53^-1 + 30 + 20*53 + 47*53^2 + 10*53^3 + 32*53^4 + 9*53^5 + 22*53^6 + 35*53^7 + 30*53^8 + O(53^9) An anomalous case where the precision drops some:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("5077a") sage: E.padic_regulator(5, 10) 5 + 5^2 + 4*5^3 + 2*5^4 + 2*5^5 + 2*5^6 + 4*5^7 + 2*5^8 + 5^9 + O(5^10) Check that answers agree over a range of precisions:: + sage: # needs sage.libs.eclib sage: max_prec = 30 # make sure we get past p^2 # long time sage: full = E.padic_regulator(5, max_prec) # long time sage: for prec in range(1, max_prec): # long time @@ -296,12 +305,14 @@ def padic_regulator(self, p, prec=20, height=None, check_hypotheses=True): A case where the generator belongs to the formal group already (:issue:`3632`):: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([37,0]) sage: E.padic_regulator(5,10) 2*5^2 + 2*5^3 + 5^4 + 5^5 + 4*5^6 + 3*5^8 + 4*5^9 + O(5^10) The result is not dependent on the model for the curve:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0,0,0,0,2^12*17]) sage: Em = E.minimal_model() sage: E.padic_regulator(7) == Em.padic_regulator(7) @@ -309,6 +320,7 @@ def padic_regulator(self, p, prec=20, height=None, check_hypotheses=True): Allow a python int as input:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a') sage: E.padic_regulator(int(5)) 5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + 5^10 + 3*5^11 + 3*5^12 + 5^13 + 4*5^14 + 5^15 + 2*5^16 + 5^17 + 2*5^18 + 4*5^19 + O(5^20) @@ -373,6 +385,7 @@ def padic_height_pairing_matrix(self, p, prec=20, height=None, check_hypotheses= EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("37a") sage: E.padic_height_pairing_matrix(5, 10) [5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + O(5^10)] @@ -495,6 +508,7 @@ def _multiply_point(E, R, P, m): Test over a range of `n` for a single curve with fairly random coefficients:: + sage: # needs sage.libs.eclib sage: R = Integers(625) sage: E = EllipticCurve([4, -11, 17, -8, -10]) sage: P = E.gens()[0] * LCM(E.tamagawa_numbers()) @@ -708,6 +722,7 @@ def padic_height(self, p, prec=20, sigma=None, check_hypotheses=True): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("37a") sage: P = E.gens()[0] sage: h = E.padic_height(5, 10) @@ -716,22 +731,26 @@ def padic_height(self, p, prec=20, sigma=None, check_hypotheses=True): An anomalous case:: + sage: # needs sage.libs.eclib sage: h = E.padic_height(53, 10) sage: h(P) 26*53^-1 + 30 + 20*53 + 47*53^2 + 10*53^3 + 32*53^4 + 9*53^5 + 22*53^6 + 35*53^7 + 30*53^8 + 17*53^9 + O(53^10) Boundary case:: + sage: # needs sage.libs.eclib sage: E.padic_height(5, 3)(P) 5 + 5^2 + O(5^3) A case that works the division polynomial code a little harder:: + sage: # needs sage.libs.eclib sage: E.padic_height(5, 10)(5*P) 5^3 + 5^4 + 5^5 + 3*5^8 + 4*5^9 + O(5^10) Check that answers agree over a range of precisions:: + sage: # needs sage.libs.eclib sage: max_prec = 30 # make sure we get past p^2 # long time sage: full = E.padic_height(5, max_prec)(P) # long time sage: for prec in range(1, max_prec): # long time @@ -739,6 +758,7 @@ def padic_height(self, p, prec=20, sigma=None, check_hypotheses=True): A supersingular prime for a curve:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a') sage: E.is_supersingular(3) True @@ -752,6 +772,7 @@ def padic_height(self, p, prec=20, sigma=None, check_hypotheses=True): A torsion point in both the good and supersingular cases:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a') sage: P = E.torsion_subgroup().gen(0).element(); P (5 : 5 : 1) @@ -764,6 +785,7 @@ def padic_height(self, p, prec=20, sigma=None, check_hypotheses=True): The result is not dependent on the model for the curve:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0,0,0,0,2^12*17]) sage: Em = E.minimal_model() sage: P = E.gens()[0] @@ -777,6 +799,7 @@ def padic_height(self, p, prec=20, sigma=None, check_hypotheses=True): Check that issue :issue:`20798` is solved:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("91b") sage: h = E.padic_height(7,10) sage: P = E.gen(0) @@ -903,6 +926,7 @@ def padic_height_via_multiply(self, p, prec=20, E2=None, check_hypotheses=True): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("37a") sage: P = E.gens()[0] sage: h = E.padic_height_via_multiply(5, 10) @@ -911,12 +935,14 @@ def padic_height_via_multiply(self, p, prec=20, E2=None, check_hypotheses=True): An anomalous case:: + sage: # needs sage.libs.eclib sage: h = E.padic_height_via_multiply(53, 10) sage: h(P) 26*53^-1 + 30 + 20*53 + 47*53^2 + 10*53^3 + 32*53^4 + 9*53^5 + 22*53^6 + 35*53^7 + 30*53^8 + 17*53^9 + O(53^10) Supply the value of E2 manually:: + sage: # needs sage.libs.eclib sage: E2 = E.padic_E2(5, 8) sage: E2 2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + O(5^8) @@ -926,11 +952,13 @@ def padic_height_via_multiply(self, p, prec=20, E2=None, check_hypotheses=True): Boundary case:: + sage: # needs sage.libs.eclib sage: E.padic_height_via_multiply(5, 3)(P) 5 + 5^2 + O(5^3) Check that answers agree over a range of precisions:: + sage: # needs sage.libs.eclib sage: max_prec = 30 # make sure we get past p^2 # long time sage: full = E.padic_height(5, max_prec)(P) # long time sage: for prec in range(2, max_prec): # long time diff --git a/src/sage/schemes/elliptic_curves/sha_tate.py b/src/sage/schemes/elliptic_curves/sha_tate.py index c67c462dbe2..328b57dabe2 100644 --- a/src/sage/schemes/elliptic_curves/sha_tate.py +++ b/src/sage/schemes/elliptic_curves/sha_tate.py @@ -34,6 +34,7 @@ EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('11a1') sage: S = E.sha() sage: S.bound_kato() @@ -47,6 +48,7 @@ sage: S.an_numerical() 1.00000000000000 + sage: # needs sage.libs.eclib sage: E = EllipticCurve('389a') sage: S = E.sha(); S Tate-Shafarevich group for the @@ -110,6 +112,7 @@ class Sha(SageObject): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('571a1') sage: E._set_gens([]) # curve has rank 0, but non-trivial Sha[2] sage: S = E.sha() @@ -124,6 +127,7 @@ class Sha(SageObject): sage: S.an_numerical() 4.00000000000000 + sage: # needs sage.libs.eclib sage: E = EllipticCurve('389a') sage: S = E.sha(); S Tate-Shafarevich group for the @@ -238,6 +242,7 @@ def an_numerical(self, prec=None, EXAMPLES:: + sage: # needs.sage.libs.eclib sage: EllipticCurve('11a').sha().an_numerical() 1.00000000000000 sage: EllipticCurve('37a').sha().an_numerical() @@ -251,18 +256,21 @@ def an_numerical(self, prec=None, A rank 4 curve:: + sage: # needs.sage.libs.eclib sage: EllipticCurve([1, -1, 0, -79, 289]).sha().an_numerical() # long time (3s on sage.math, 2011) 1.00000000000000 A rank 5 curve:: - sage: EllipticCurve([0, 0, 1, -79, 342]).sha().an_numerical(prec=10, proof=False) # long time (22s on sage.math, 2011) + sage: # long time, needs.sage.libs.eclib + sage: EllipticCurve([0, 0, 1, -79, 342]).sha().an_numerical(prec=10, proof=False) 1.0 See :issue:`1115`:: + sage: # long time, needs.sage.libs.eclib sage: sha = EllipticCurve('37a1').sha() - sage: [sha.an_numerical(prec) for prec in range(40,100,10)] # long time (3s on sage.math, 2013) + sage: [sha.an_numerical(prec) for prec in range(40,100,10)] [1.0000000000, 1.0000000000000, 1.0000000000000000, @@ -324,6 +332,7 @@ def an(self, use_database=False, descent_second_limit=12): If you come across the following error:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0, 0, 1, -34874, -2506691]) sage: E.sha().an() Traceback (most recent call last): @@ -335,11 +344,13 @@ def an(self, use_database=False, descent_second_limit=12): You can increase the ``descent_second_limit`` (in the above example, set to the default, 12) option to try again:: + sage: # needs sage.libs.eclib sage: E.sha().an(descent_second_limit=16) # long time (2s on sage.math, 2011) 1 EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0, -1, 1, -10, -20]) # 11A = X_0(11) sage: E.sha().an() 1 @@ -347,6 +358,7 @@ def an(self, use_database=False, descent_second_limit=12): sage: E.sha().an() 1 + sage: # needs sage.libs.eclib sage: EllipticCurve('14a4').sha().an() 1 sage: EllipticCurve('14a4').sha().an(use_database=True) # will be faster if you have large Cremona database installed @@ -354,12 +366,14 @@ def an(self, use_database=False, descent_second_limit=12): The smallest conductor curve with nontrivial `Sha`:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([1,1,1,-352,-2689]) # 66b3 sage: E.sha().an() 4 The four optimal quotients with nontrivial `Sha` and conductor <= 1000:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0, -1, 1, -929, -10595]) # 571A sage: E.sha().an() 4 @@ -375,6 +389,7 @@ def an(self, use_database=False, descent_second_limit=12): The smallest conductor curve of rank > 1:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0, 1, 1, -2, 0]) # 389A (rank 2) sage: E.sha().an() 1.00000000000000 @@ -382,10 +397,12 @@ def an(self, use_database=False, descent_second_limit=12): The following are examples that require computation of the Mordell- Weil group and regulator:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0, 0, 1, -1, 0]) # 37A (rank 1) sage: E.sha().an() 1 + sage: # needs sage.libs.eclib sage: E = EllipticCurve("1610f3") sage: E.sha().an() 4 @@ -393,6 +410,7 @@ def an(self, use_database=False, descent_second_limit=12): In this case the input curve is not minimal, and if this function did not transform it to be minimal, it would give nonsense:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([0, -432*6^2]) sage: E.sha().an() 1 @@ -400,6 +418,7 @@ def an(self, use_database=False, descent_second_limit=12): See :issue:`10096`: this used to give the wrong result 6.0000 before since the minimal model was not used:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([1215*1216, 0]) # non-minimal model sage: E.sha().an() # long time (2s on sage.math, 2011) 1.00000000000000 @@ -485,6 +504,7 @@ def an_padic(self, p, prec=0, use_twists=True): Good ordinary examples:: + sage: # needs sage.libs.eclib sage: EllipticCurve('11a1').sha().an_padic(5) # rank 0 1 + O(5^22) sage: EllipticCurve('43a1').sha().an_padic(5) # rank 1 @@ -500,6 +520,7 @@ def an_padic(self, p, prec=0, use_twists=True): Exceptional cases:: + sage: # needs sage.libs.eclib sage: EllipticCurve('11a1').sha().an_padic(11) # rank 0 1 + O(11^22) sage: EllipticCurve('130a1').sha().an_padic(5) # rank 1 @@ -507,16 +528,19 @@ def an_padic(self, p, prec=0, use_twists=True): Non-split, but rank 0 case (:issue:`7331`):: + sage: # needs sage.libs.eclib sage: EllipticCurve('270b1').sha().an_padic(5) # rank 0, long time (2s on sage.math, 2011) 1 + O(5^22) The output has the correct sign:: + sage: # needs sage.libs.eclib sage: EllipticCurve('123a1').sha().an_padic(41) # rank 1, long time (3s on sage.math, 2011) 1 + O(41) Supersingular cases:: + sage: # needs sage.libs.eclib sage: EllipticCurve('34a1').sha().an_padic(5) # rank 0 1 + O(5^22) sage: EllipticCurve('53a1').sha().an_padic(5) # rank 1, long time (11s on sage.math, 2011) @@ -524,6 +548,7 @@ def an_padic(self, p, prec=0, use_twists=True): Cases that use a twist to a lower conductor:: + sage: # needs sage.libs.eclib sage: EllipticCurve('99a1').sha().an_padic(5) 1 + O(5) sage: EllipticCurve('240d3').sha().an_padic(5) # sha has 4 elements here @@ -535,6 +560,7 @@ def an_padic(self, p, prec=0, use_twists=True): Test for :issue:`15737`:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve([-100,0]) sage: s = E.sha() sage: s.an_padic(13) @@ -734,6 +760,7 @@ def p_primary_order(self, p): EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve("389a1") # rank 2 sage: E.sha().p_primary_order(5) 0 @@ -811,6 +838,7 @@ def p_primary_bound(self, p): EXAMPLES:: + sage: # needs sage.libs.eclib sage: e = EllipticCurve('11a3') sage: e.sha().p_primary_bound(3) 0 @@ -823,6 +851,7 @@ def p_primary_bound(self, p): sage: e.sha().p_primary_bound(13) 0 + sage: # needs sage.libs.eclib sage: e = EllipticCurve('389a1') sage: e.sha().p_primary_bound(5) 0 @@ -833,13 +862,15 @@ def p_primary_bound(self, p): sage: e.sha().p_primary_bound(13) 0 + sage: # long time, needs sage.libs.eclib sage: e = EllipticCurve('858k2') - sage: e.sha().p_primary_bound(3) # long time (10s on sage.math, 2011) + sage: e.sha().p_primary_bound(3) 0 Some checks for :issue:`6406` and :issue:`16959`:: - sage: e.sha().p_primary_bound(7) # long time + sage: # long time, needs sage.libs.eclib + sage: e.sha().p_primary_bound(7) 2 sage: E = EllipticCurve('608b1') @@ -850,11 +881,13 @@ def p_primary_bound(self, p): Current knowledge about Euler systems does not provide an upper bound in this case. Try an_padic for a conjectural bound. - sage: E.sha().an_padic(5) # long time + sage: # long time, needs sage.libs.eclib + sage: E.sha().an_padic(5) 1 + O(5^22) + sage: # long time, needs sage.libs.eclib sage: E = EllipticCurve("5040bi1") - sage: E.sha().p_primary_bound(5) # long time + sage: E.sha().p_primary_bound(5) 0 """ p = Integer(p) @@ -957,6 +990,7 @@ def bound_kolyvagin(self, D=0, regulator=None, EXAMPLES:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('37a') sage: E.sha().bound_kolyvagin() ([2], 1) @@ -968,6 +1002,7 @@ def bound_kolyvagin(self, D=0, regulator=None, We get no information when the curve has rank 2.:: + sage: # needs sage.libs.eclib sage: E = EllipticCurve('389a') sage: E.sha().bound_kolyvagin() (0, 0) @@ -1148,6 +1183,7 @@ def bound(self): EXAMPLES:: + sage: # needs sage.libs.eclib sage: EllipticCurve('37a').sha().bound() ([2], 1) """ diff --git a/src/sage/tests/benchmark.py b/src/sage/tests/benchmark.py index 7f5962df896..1852700c47a 100644 --- a/src/sage/tests/benchmark.py +++ b/src/sage/tests/benchmark.py @@ -1635,7 +1635,7 @@ def sage(self): sage: from sage.tests.benchmark import EllipticCurveMW sage: B = EllipticCurveMW([1,2,3,4,5]) - sage: isinstance(B.sage()[1], float) + sage: isinstance(B.sage()[1], float) # needs sage.libs.eclib True """ E = EllipticCurve(self.ainvs) diff --git a/src/sage/tests/book_stein_ent.py b/src/sage/tests/book_stein_ent.py index 2e942934b9f..ce5fd8c004b 100644 --- a/src/sage/tests/book_stein_ent.py +++ b/src/sage/tests/book_stein_ent.py @@ -628,20 +628,20 @@ 2 sage: r([-112, 400]) 3 -sage: r([-102627, 12560670]) +sage: r([-102627, 12560670]) # needs sage.libs.eclib 4 sage: def cong(n): ....: G = EllipticCurve([-n^2,0]).gens() ....: if len(G) == 0: return False ....: x,y,_ = G[0] ....: return ((n^2-x^2)/y,-2*x*n/y,(n^2+x^2)/y) -sage: cong(6) +sage: cong(6) # needs sage.libs.eclib (3, 4, 5) -sage: cong(5) +sage: cong(5) # needs sage.libs.eclib (3/2, 20/3, 41/6) -sage: cong(1) +sage: cong(1) # needs sage.libs.eclib False -sage: cong(13) +sage: cong(13) # needs sage.libs.eclib (323/30, 780/323, 106921/9690) sage: (323/30 * 780/323)/2 13