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# -*- coding: utf-8 -*- r""" Elliptic curves over number fields
An elliptic curve `E` over a number field `K` can be given by a Weierstrass equation whose coefficients lie in `K` or by using ``base_extend`` on an elliptic curve defined over a subfield.
One major difference to elliptic curves over `\QQ` is that there might not exist a global minimal equation over `K`, when `K` does not have class number one. Another difference is the lack of understanding of modularity for general elliptic curves over general number fields.
Currently Sage can obtain local information about `E/K_v` for finite places `v`, it has an interface to Denis Simon's script for 2-descent, it can compute the torsion subgroup of the Mordell-Weil group `E(K)`, and it can work with isogenies defined over `K`.
EXAMPLES::
sage: K.<i> = NumberField(x^2+1) sage: E = EllipticCurve([0,4+i]) sage: E.discriminant() -3456*i - 6480 sage: P= E([i,2]) sage: P+P (-2*i + 9/16 : -9/4*i - 101/64 : 1)
::
sage: E.has_good_reduction(2+i) True sage: E.local_data(4+i) Local data at Fractional ideal (i + 4): Reduction type: bad additive Local minimal model: Elliptic Curve defined by y^2 = x^3 + (i+4) over Number Field in i with defining polynomial x^2 + 1 Minimal discriminant valuation: 2 Conductor exponent: 2 Kodaira Symbol: II Tamagawa Number: 1 sage: E.tamagawa_product_bsd() 1
::
sage: E.simon_two_descent() (1, 1, [(i : 2 : 1)])
::
sage: E.torsion_order() 1
::
sage: E.isogenies_prime_degree(3) [Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + (i+4) over Number Field in i with defining polynomial x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + (-27*i-108) over Number Field in i with defining polynomial x^2 + 1]
AUTHORS:
- Robert Bradshaw 2007
- John Cremona
- Chris Wuthrich
REFERENCE:
- [Sil] Silverman, Joseph H. The arithmetic of elliptic curves. Second edition. Graduate Texts in Mathematics, 106. Springer, 2009.
- [Sil2] Silverman, Joseph H. Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, 151. Springer, 1994. """ from __future__ import absolute_import
#***************************************************************************** # Copyright (C) 2007 Robert Bradshaw <robertwb@math.washington.edu> # William Stein <wstein@gmail.com> # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
from .ell_field import EllipticCurve_field from .ell_generic import is_EllipticCurve from .ell_point import EllipticCurvePoint_number_field from .constructor import EllipticCurve from sage.rings.all import PolynomialRing, ZZ, QQ, RealField, Integer from sage.misc.all import cached_method, verbose, prod, union, flatten from six import reraise as raise_
class EllipticCurve_number_field(EllipticCurve_field): r""" Elliptic curve over a number field.
EXAMPLES::
sage: K.<i>=NumberField(x^2+1) sage: EllipticCurve([i, i - 1, i + 1, 24*i + 15, 14*i + 35]) Elliptic Curve defined by y^2 + i*x*y + (i+1)*y = x^3 + (i-1)*x^2 + (24*i+15)*x + (14*i+35) over Number Field in i with defining polynomial x^2 + 1 """ def __init__(self, K, ainvs): r""" EXAMPLES:
A curve from the database of curves over `\QQ`, but over a larger field:
sage: K.<i>=NumberField(x^2+1) sage: EllipticCurve(K,'389a1') Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-2)*x over Number Field in i with defining polynomial x^2 + 1
Making the field of definition explicitly larger::
sage: EllipticCurve(K,[0,-1,1,0,0]) Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 over Number Field in i with defining polynomial x^2 + 1
"""
_point = EllipticCurvePoint_number_field
def base_extend(self, R): """ Return the base extension of ``self`` to `R`.
EXAMPLES::
sage: E = EllipticCurve('11a3') sage: K = QuadraticField(-5, 'a') sage: E.base_extend(K) Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 over Number Field in a with defining polynomial x^2 + 5
Check that non-torsion points are remembered when extending the base field (see :trac:`16034`)::
sage: E = EllipticCurve([1, 0, 1, -1751, -31352]) sage: K.<d> = QuadraticField(5) sage: E.gens() [(52 : 111 : 1)] sage: EK = E.base_extend(K) sage: EK.gens() [(52 : 111 : 1)]
"""
def simon_two_descent(self, verbose=0, lim1=2, lim3=4, limtriv=2, maxprob=20, limbigprime=30, known_points=None): r""" Return lower and upper bounds on the rank of the Mordell-Weil group `E(K)` and a list of points.
This method is used internally by the :meth:`~rank`, :meth:`~rank_bounds` and :meth:`~gens` methods.
INPUT:
- ``self`` -- an elliptic curve `E` over a number field `K`
- ``verbose`` -- 0, 1, 2, or 3 (default: 0), the verbosity level
- ``lim1`` -- (default: 2) limit on trivial points on quartics
- ``lim3`` -- (default: 4) limit on points on ELS quartics
- ``limtriv`` -- (default: 2) limit on trivial points on `E`
- ``maxprob`` -- (default: 20)
- ``limbigprime`` -- (default: 30) to distinguish between small and large prime numbers. Use probabilistic tests for large primes. If 0, don't use probabilistic tests.
- ``known_points`` -- (default: None) list of known points on the curve
OUTPUT: a triple ``(lower, upper, list)`` consisting of
- ``lower`` (integer) -- lower bound on the rank
- ``upper`` (integer) -- upper bound on the rank
- ``list`` -- list of points in `E(K)`
The integer ``upper`` is in fact an upper bound on the dimension of the 2-Selmer group, hence on the dimension of `E(K)/2E(K)`. It is equal to the dimension of the 2-Selmer group except possibly if `E(K)[2]` has dimension 1. In that case, ``upper`` may exceed the dimension of the 2-Selmer group by an even number, due to the fact that the algorithm does not perform a second descent.
.. note::
For non-quadratic number fields, this code does return, but it takes a long time.
ALGORITHM:
Uses Denis Simon's PARI/GP scripts from http://www.math.unicaen.fr/~simon/.
EXAMPLES::
sage: K.<a> = NumberField(x^2 + 23, 'a') sage: E = EllipticCurve(K, '37') sage: E == loads(dumps(E)) True sage: E.simon_two_descent() (2, 2, [(0 : 0 : 1), (1/8*a + 5/8 : -3/16*a - 7/16 : 1)]) sage: E.simon_two_descent(lim1=3, lim3=20, limtriv=5, maxprob=7, limbigprime=10) (2, 2, [(-1 : 0 : 1), (-1/8*a + 5/8 : -3/16*a - 9/16 : 1)])
::
sage: K.<a> = NumberField(x^2 + 7, 'a') sage: E = EllipticCurve(K, [0,0,0,1,a]); E Elliptic Curve defined by y^2 = x^3 + x + a over Number Field in a with defining polynomial x^2 + 7
sage: v = E.simon_two_descent(verbose=1); v elliptic curve: Y^2 = x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7) Trivial points on the curve = [[1, 1, 0], [Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7), 1]] #S(E/K)[2] = 2 #E(K)/2E(K) = 2 #III(E/K)[2] = 1 rank(E/K) = 1 listpoints = [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7), 1]] (1, 1, [(1/2*a + 3/2 : -a - 2 : 1)])
sage: v = E.simon_two_descent(verbose=2) K = bnfinit(y^2 + 7); a = Mod(y,K.pol); bnfellrank(K, [0, 0, 0, 1, a], [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)]]); elliptic curve: Y^2 = x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7) A = Mod(0, y^2 + 7) B = Mod(1, y^2 + 7) C = Mod(y, y^2 + 7) <BLANKLINE> Computing L(S,2) L(S,2) = [Mod(Mod(-1/2*y + 1/2, y^2 + 7)*x^2 + Mod(-1/2*y - 1/2, y^2 + 7)*x + Mod(-y - 1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(Mod(-1, y^2 + 7)*x^2 + Mod(-1/2*y - 1/2, y^2 + 7)*x + Mod(1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(-1, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(x^2 + 2, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(Mod(1, y^2 + 7)*x + Mod(1/2*y + 3/2, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(Mod(1, y^2 + 7)*x + Mod(1/2*y - 3/2, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))] <BLANKLINE> Computing the Selmer group #LS2gen = 2 LS2gen = [Mod(Mod(-1/2*y + 1/2, y^2 + 7)*x^2 + Mod(-1/2*y - 1/2, y^2 + 7)*x + Mod(-y - 1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(Mod(1, y^2 + 7)*x^2 + Mod(1/2*y + 1/2, y^2 + 7)*x + Mod(-1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))] Search for trivial points on the curve Trivial points on the curve = [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)], [1, 1, 0], [Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7), 1]] zc = Mod(Mod(-1/2*y + 1/2, y^2 + 7)*x^2 + Mod(-1/2*y - 1/2, y^2 + 7)*x + Mod(-y - 1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)) Hilbert symbol (Mod(1, y^2 + 7),Mod(-2*y + 2, y^2 + 7)) = sol of quadratic equation = [1, 1, 0]~ zc*z1^2 = Mod(Mod(4, y^2 + 7)*x + Mod(-2*y + 6, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)) quartic: (-1)*Y^2 = x^4 + (3*y - 9)*x^2 + (-8*y + 16)*x + (9/2*y - 11/2) reduced: Y^2 = -x^4 + (-3*y + 9)*x^2 + (-8*y + 16)*x + (-9/2*y + 11/2) not ELS at [2, [0, 1]~, 1, 1, [1, -2; 1, 0]] zc = Mod(Mod(1, y^2 + 7)*x^2 + Mod(1/2*y + 1/2, y^2 + 7)*x + Mod(-1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)) comes from the trivial point [Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)] m1 = 1 m2 = 1 #S(E/K)[2] = 2 #E(K)/2E(K) = 2 #III(E/K)[2] = 1 rank(E/K) = 1 listpoints = [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)]] v = [1, 1, [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)]]] sage: v (1, 1, [(1/2*a + 3/2 : -a - 2 : 1)])
A curve with 2-torsion::
sage: K.<a> = NumberField(x^2 + 7) sage: E = EllipticCurve(K, '15a') sage: E.simon_two_descent() # long time (3s on sage.math, 2013), points can vary (1, 3, [...])
Check that the bug reported in :trac:`15483` is fixed::
sage: K.<s> = QuadraticField(229) sage: c4 = 2173 - 235*(1 - s)/2 sage: c6 = -124369 + 15988*(1 - s)/2 sage: E = EllipticCurve([-c4/48, -c6/864]) sage: E.simon_two_descent() (0, 0, [])
sage: R.<t> = QQ[] sage: L.<g> = NumberField(t^3 - 9*t^2 + 13*t - 4) sage: E1 = EllipticCurve(L,[1-g*(g-1),-g^2*(g-1),-g^2*(g-1),0,0]) sage: E1.rank() # long time (about 5 s) 0
sage: K = CyclotomicField(43).subfields(3)[0][0] sage: E = EllipticCurve(K, '37') sage: E.simon_two_descent() # long time (4s on sage.math, 2013) (3, 3, [(0 : 0 : 1), (-1/2*zeta43_0^2 - 1/2*zeta43_0 + 7 : -3/2*zeta43_0^2 - 5/2*zeta43_0 + 18 : 1)]) """
# We deliberately do not use known_points as a key in the # following caching code, so that calling E.gens() a second # time (when known_points may have increased) will not cause # another execution of simon_two_descent.
lim1=lim1, lim3=lim3, limtriv=limtriv, maxprob=maxprob, limbigprime=limbigprime, known_points=known_points) if P not in self._known_points])
def division_field(self, p, names, map=False, **kwds): """ Given an elliptic curve over a number field `F` and a prime number `p`, construct the field `F(E[p])`.
INPUT:
- ``p`` -- a prime number (an element of `\ZZ`)
- ``names`` -- a variable name for the number field
- ``map`` -- (default: ``False``) also return an embedding of the :meth:`base_field` into the resulting field.
- ``kwds`` -- additional keywords passed to :func:`sage.rings.number_field.splitting_field.splitting_field`.
OUTPUT:
If ``map`` is ``False``, the division field as an absolute number field. If ``map`` is ``True``, a tuple ``(K, phi)`` where ``phi`` is an embedding of the base field in the division field ``K``.
.. WARNING::
This takes a very long time when the degree of the division field is large (e.g. when `p` is large or when the Galois representation is surjective). The ``simplify`` flag also has a big influence on the running time: sometimes ``simplify=False`` is faster, sometimes ``simplify=True`` (the default) is faster.
EXAMPLES:
The 2-division field is the same as the splitting field of the 2-division polynomial (therefore, it has degree 1, 2, 3 or 6)::
sage: E = EllipticCurve('15a1') sage: K.<b> = E.division_field(2); K Number Field in b with defining polynomial x sage: E = EllipticCurve('14a1') sage: K.<b> = E.division_field(2); K Number Field in b with defining polynomial x^2 + 5*x + 92 sage: E = EllipticCurve('196b1') sage: K.<b> = E.division_field(2); K Number Field in b with defining polynomial x^3 + x^2 - 114*x - 127 sage: E = EllipticCurve('19a1') sage: K.<b> = E.division_field(2); K Number Field in b with defining polynomial x^6 + 10*x^5 + 24*x^4 - 212*x^3 + 1364*x^2 + 24072*x + 104292
For odd primes `p`, the division field is either the splitting field of the `p`-division polynomial, or a quadratic extension of it. ::
sage: E = EllipticCurve('50a1') sage: F.<a> = E.division_polynomial(3).splitting_field(simplify_all=True); F Number Field in a with defining polynomial x^6 - 3*x^5 + 4*x^4 - 3*x^3 - 2*x^2 + 3*x + 3 sage: K.<b> = E.division_field(3, simplify_all=True); K Number Field in b with defining polynomial x^6 - 3*x^5 + 4*x^4 - 3*x^3 - 2*x^2 + 3*x + 3
If we take any quadratic twist, the splitting field of the 3-division polynomial remains the same, but the 3-division field becomes a quadratic extension::
sage: E = E.quadratic_twist(5) # 50b3 sage: F.<a> = E.division_polynomial(3).splitting_field(simplify_all=True); F Number Field in a with defining polynomial x^6 - 3*x^5 + 4*x^4 - 3*x^3 - 2*x^2 + 3*x + 3 sage: K.<b> = E.division_field(3, simplify_all=True); K Number Field in b with defining polynomial x^12 - 3*x^11 + 8*x^10 - 15*x^9 + 30*x^8 - 63*x^7 + 109*x^6 - 144*x^5 + 150*x^4 - 120*x^3 + 68*x^2 - 24*x + 4
Try another quadratic twist, this time over a subfield of `F`::
sage: G.<c>,_,_ = F.subfields(3)[0] sage: E = E.base_extend(G).quadratic_twist(c); E Elliptic Curve defined by y^2 = x^3 + 5*a0*x^2 + (-200*a0^2)*x + (-42000*a0^2+42000*a0+126000) over Number Field in a0 with defining polynomial x^3 - 3*x^2 + 3*x + 9 sage: K.<b> = E.division_field(3, simplify_all=True); K Number Field in b with defining polynomial x^12 - 10*x^10 + 55*x^8 - 60*x^6 + 75*x^4 + 1350*x^2 + 2025
Some higher-degree examples::
sage: E = EllipticCurve('11a1') sage: K.<b> = E.division_field(2); K Number Field in b with defining polynomial x^6 + 2*x^5 - 48*x^4 - 436*x^3 + 1668*x^2 + 28792*x + 73844 sage: K.<b> = E.division_field(3); K # long time (3s on sage.math, 2014) Number Field in b with defining polynomial x^48 ... sage: K.<b> = E.division_field(5); K Number Field in b with defining polynomial x^4 - x^3 + x^2 - x + 1 sage: E.division_field(5, 'b', simplify=False) Number Field in b with defining polynomial x^4 + x^3 + 11*x^2 + 41*x + 101 sage: E.base_extend(K).torsion_subgroup() # long time (2s on sage.math, 2014) Torsion Subgroup isomorphic to Z/5 + Z/5 associated to the Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in b with defining polynomial x^4 - x^3 + x^2 - x + 1
sage: E = EllipticCurve('27a1') sage: K.<b> = E.division_field(3); K Number Field in b with defining polynomial x^2 + 3*x + 9 sage: K.<b> = E.division_field(2); K Number Field in b with defining polynomial x^6 + 6*x^5 + 24*x^4 - 52*x^3 - 228*x^2 + 744*x + 3844 sage: K.<b> = E.division_field(2, simplify_all=True); K Number Field in b with defining polynomial x^6 - 3*x^5 + 5*x^3 - 3*x + 1 sage: K.<b> = E.division_field(5); K # long time (4s on sage.math, 2014) Number Field in b with defining polynomial x^48 ... sage: K.<b> = E.division_field(7); K # long time (8s on sage.math, 2014) Number Field in b with defining polynomial x^72 ...
Over a number field::
sage: R.<x> = PolynomialRing(QQ) sage: K.<i> = NumberField(x^2 + 1) sage: E = EllipticCurve([0,0,0,0,i]) sage: L.<b> = E.division_field(2); L Number Field in b with defining polynomial x^4 - x^2 + 1 sage: L.<b>, phi = E.division_field(2, map=True); phi Ring morphism: From: Number Field in i with defining polynomial x^2 + 1 To: Number Field in b with defining polynomial x^4 - x^2 + 1 Defn: i |--> -b^3 sage: L.<b>, phi = E.division_field(3, map=True) sage: L Number Field in b with defining polynomial x^24 - 6*x^22 - 12*x^21 - 21*x^20 + 216*x^19 + 48*x^18 + 804*x^17 + 1194*x^16 - 13488*x^15 + 21222*x^14 + 44196*x^13 - 47977*x^12 - 102888*x^11 + 173424*x^10 - 172308*x^9 + 302046*x^8 + 252864*x^7 - 931182*x^6 + 180300*x^5 + 879567*x^4 - 415896*x^3 + 1941012*x^2 + 650220*x + 443089 sage: phi Ring morphism: From: Number Field in i with defining polynomial x^2 + 1 To: Number Field in b with defining polynomial x^24 ... Defn: i |--> -215621657062634529/183360797284413355040732*b^23 ...
AUTHORS:
- Jeroen Demeyer (2014-01-06): :trac:`11905`, use ``splitting_field`` method, moved from ``gal_reps.py``, make it work over number fields. """ raise ValueError("p must be a prime number")
# For p = 2, the division field is the splitting field of # the division polynomial.
# Compute splitting field of X-coordinates. # The Galois group of the division field is a subgroup of GL(2,p). # The Galois group of the X-coordinates is a subgroup of GL(2,p)/{-1,+1}. # We need the map to change the elliptic curve invariants to K.
# THEOREM (Cremona, http://trac.sagemath.org/ticket/11905#comment:21). # Let K be a field, E an elliptic curve over K and p an odd # prime number. Assume that K contains all roots of the # p-division polynomial of E. Then either K contains all # p-torsion points on E, or it doesn't contain any p-torsion # point. # # PROOF. Let G be the absolute Galois group of K (every element # in it fixes all elements of K). For any p-torsion point P # over the algebraic closure and any sigma in G, we must have # either sigma(P) = P or sigma(P) = -P (since K contains the # X-coordinate of P). Now assume that K does not contain all # p-torsion points. Then there exists a point P1 and a sigma in # G such that sigma(P1) = -P1. Now take a different p-torsion # point P2. Since sigma(P2) must be P2 or -P2 and # sigma(P1+P2) = sigma(P1)+sigma(P2) = sigma(P1)-P2 must # be P1+P2 or -(P1+P2), it follows that sigma(P2) = -sigma(P2). # Therefore, K cannot contain any p-torsion point. # # This implies that it suffices to adjoin the Y-coordinate # of just one point.
# First factor f over F and then compute a root X of f over K.
# Polynomial defining the corresponding Y-coordinate else:
def height_pairing_matrix(self, points=None, precision=None): r""" Returns the height pairing matrix of the given points.
INPUT:
- points -- either a list of points, which must be on this curve, or (default) None, in which case self.gens() will be used.
- precision -- number of bits of precision of result (default: None, for default RealField precision)
EXAMPLES::
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: E=EllipticCurve('5077a1') sage: E.height_pairing_matrix([E.lift_x(x) for x in [-2,-7/4,1]], precision=100) [ 1.3685725053539301120518194471 -1.3095767070865761992624519454 -0.63486715783715592064475542573] [ -1.3095767070865761992624519454 2.7173593928122930896610589220 1.0998184305667292139777571432] [-0.63486715783715592064475542573 1.0998184305667292139777571432 0.66820516565192793503314205089]
sage: E = EllipticCurve('389a1') sage: E = EllipticCurve('389a1') sage: P,Q = E.point([-1,1,1]),E.point([0,-1,1]) sage: E.height_pairing_matrix([P,Q]) [0.686667083305587 0.268478098806726] [0.268478098806726 0.327000773651605]
Over a number field::
sage: x = polygen(QQ) sage: K.<t> = NumberField(x^2+47) sage: EK = E.base_extend(K) sage: EK.height_pairing_matrix([EK(P),EK(Q)]) [0.686667083305587 0.268478098806726] [0.268478098806726 0.327000773651605]
::
sage: K.<i> = QuadraticField(-1) sage: E = EllipticCurve([0,0,0,i,i]) sage: P = E(-9+4*i,-18-25*i) sage: Q = E(i,-i) sage: E.height_pairing_matrix([P,Q]) [ 2.16941934493768 -0.870059380421505] [-0.870059380421505 0.424585837470709] sage: E.regulator_of_points([P,Q]) 0.164101403936070 """ else:
else:
def regulator_of_points(self, points=[], precision=None): """ Returns the regulator of the given points on this curve.
INPUT:
- ``points`` -(default: empty list) a list of points on this curve
- ``precision`` - int or None (default: None): the precision in bits of the result (default real precision if None)
EXAMPLES::
sage: E = EllipticCurve('37a1') sage: P = E(0,0) sage: Q = E(1,0) sage: E.regulator_of_points([P,Q]) 0.000000000000000 sage: 2*P == Q True
::
sage: E = EllipticCurve('5077a1') sage: points = [E.lift_x(x) for x in [-2,-7/4,1]] sage: E.regulator_of_points(points) 0.417143558758384 sage: E.regulator_of_points(points,precision=100) 0.41714355875838396981711954462
::
sage: E = EllipticCurve('389a') sage: E.regulator_of_points() 1.00000000000000 sage: points = [P,Q] = [E(-1,1),E(0,-1)] sage: E.regulator_of_points(points) 0.152460177943144 sage: E.regulator_of_points(points, precision=100) 0.15246017794314375162432475705 sage: E.regulator_of_points(points, precision=200) 0.15246017794314375162432475704945582324372707748663081784028 sage: E.regulator_of_points(points, precision=300) 0.152460177943143751624324757049455823243727077486630817840280980046053225683562463604114816
Examples over number fields::
sage: K.<a> = QuadraticField(97) sage: E = EllipticCurve(K,[1,1]) sage: P = E(0,1) sage: P.height() 0.476223106404866 sage: E.regulator_of_points([P]) 0.476223106404866
::
sage: E = EllipticCurve('11a1') sage: x = polygen(QQ) sage: K.<t> = NumberField(x^2+47) sage: EK = E.base_extend(K) sage: T = EK(5,5) sage: T.order() 5 sage: P = EK(-2, -1/2*t - 1/2) sage: P.order() +Infinity sage: EK.regulator_of_points([P,T]) # random very small output -1.23259516440783e-32 sage: EK.regulator_of_points([P,T]).abs() < 1e-30 True
::
sage: E = EllipticCurve('389a1') sage: P,Q = E.gens() sage: E.regulator_of_points([P,Q]) 0.152460177943144 sage: K.<t> = NumberField(x^2+47) sage: EK = E.base_extend(K) sage: EK.regulator_of_points([EK(P),EK(Q)]) 0.152460177943144
::
sage: K.<i> = QuadraticField(-1) sage: E = EllipticCurve([0,0,0,i,i]) sage: P = E(-9+4*i,-18-25*i) sage: Q = E(i,-i) sage: E.height_pairing_matrix([P,Q]) [ 2.16941934493768 -0.870059380421505] [-0.870059380421505 0.424585837470709] sage: E.regulator_of_points([P,Q]) 0.164101403936070
""" points = []
def is_local_integral_model(self,*P): r""" Tests if self is integral at the prime ideal `P`, or at all the primes if `P` is a list or tuple.
INPUT:
- ``*P`` -- a prime ideal, or a list or tuple of primes.
EXAMPLES::
sage: K.<i> = NumberField(x^2+1) sage: P1,P2 = K.primes_above(5) sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5]) sage: E.is_local_integral_model(P1,P2) False sage: Emin = E.local_integral_model(P1,P2) sage: Emin.is_local_integral_model(P1,P2) True """
def local_integral_model(self,*P): r""" Return a model of self which is integral at the prime ideal `P`.
.. note::
The integrality at other primes is not affected, even if `P` is non-principal.
INPUT:
- ``*P`` -- a prime ideal, or a list or tuple of primes.
EXAMPLES::
sage: K.<i> = NumberField(x^2+1) sage: P1,P2 = K.primes_above(5) sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5]) sage: E.local_integral_model((P1,P2)) Elliptic Curve defined by y^2 + (-i)*x*y + (-25*i)*y = x^3 + 5*i*x^2 + 125*i*x + 3125*i over Number Field in i with defining polynomial x^2 + 1 """
def is_global_integral_model(self): r""" Return whether ``self`` is integral at all primes.
EXAMPLES::
sage: K.<i> = NumberField(x^2+1) sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5]) sage: P1,P2 = K.primes_above(5) sage: Emin = E.global_integral_model() sage: Emin.is_global_integral_model() True """
def global_integral_model(self): r""" Return a model of self which is integral at all primes.
EXAMPLES::
sage: K.<i> = NumberField(x^2+1) sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5]) sage: P1,P2 = K.primes_above(5) sage: E.global_integral_model() Elliptic Curve defined by y^2 + (-i)*x*y + (-25*i)*y = x^3 + 5*i*x^2 + 125*i*x + 3125*i over Number Field in i with defining polynomial x^2 + 1
:trac:`7935`::
sage: K.<a> = NumberField(x^2-38) sage: E = EllipticCurve([a,1/2]) sage: E.global_integral_model() Elliptic Curve defined by y^2 = x^3 + 1444*a*x + 27436 over Number Field in a with defining polynomial x^2 - 38
:trac:`9266`::
sage: K.<s> = NumberField(x^2-5) sage: w = (1+s)/2 sage: E = EllipticCurve(K,[2,w]) sage: E.global_integral_model() Elliptic Curve defined by y^2 = x^3 + 2*x + (1/2*s+1/2) over Number Field in s with defining polynomial x^2 - 5
:trac:`12151`::
sage: K.<v> = NumberField(x^2 + 161*x - 150) sage: E = EllipticCurve([25105/216*v - 3839/36, 634768555/7776*v - 98002625/1296, 634768555/7776*v - 98002625/1296, 0, 0]) sage: E.global_integral_model() Elliptic Curve defined by y^2 + (2094779518028859*v-1940492905300351)*x*y + (477997268472544193101178234454165304071127500*v-442791377441346852919930773849502871958097500)*y = x^3 + (26519784690047674853185542622500*v-24566525306469707225840460652500)*x^2 over Number Field in v with defining polynomial x^2 + 161*x - 150
:trac:`14476`::
sage: R.<t> = QQ[] sage: K.<g> = NumberField(t^4 - t^3 - 3*t^2 - t + 1) sage: E = EllipticCurve([ -43/625*g^3 + 14/625*g^2 - 4/625*g + 706/625, -4862/78125*g^3 - 4074/78125*g^2 - 711/78125*g + 10304/78125, -4862/78125*g^3 - 4074/78125*g^2 - 711/78125*g + 10304/78125, 0,0]) sage: E.global_integral_model() Elliptic Curve defined by y^2 + (15*g^3-48*g-42)*x*y + (-111510*g^3-162162*g^2-44145*g+37638)*y = x^3 + (-954*g^3-1134*g^2+81*g+576)*x^2 over Number Field in g with defining polynomial t^4 - t^3 - 3*t^2 - t + 1
"""
integral_model = global_integral_model
def _reduce_model(self): r""" Returns a reduced model for this elliptic curve.
Transforms the elliptic curve to a model which is optimally scaled with respect to units and in which `a_1`, `a_2`, `a_3` are reduced modulo 2, 3, 2 respectively.
.. note::
This only works on integral models, i.e. it requires that `a_1`, `a_2` and `a_3` lie in the ring of integers of the base field.
EXAMPLES::
sage: K.<a>=NumberField(x^2-38) sage: E=EllipticCurve([a, -5*a + 19, -39*a + 237, 368258520200522046806318224*a - 2270097978636731786720858047, 8456608930180227786550494643437985949781*a - 52130038506835491453281450568107193773505]) sage: E.ainvs() (a, -5*a + 19, -39*a + 237, 368258520200522046806318224*a - 2270097978636731786720858047, 8456608930180227786550494643437985949781*a - 52130038506835491453281450568107193773505) sage: E._reduce_model().ainvs() (a, a + 1, a + 1, 368258520200522046806318444*a - 2270097978636731786720859345, 8456608930173478039472018047583706316424*a - 52130038506793883217874390501829588391299) sage: EllipticCurve([101,202,303,404,505])._reduce_model().ainvs() (1, 1, 0, -2509254, 1528863051) sage: EllipticCurve([-101,-202,-303,-404,-505])._reduce_model().ainvs() (1, -1, 0, -1823195, 947995262)
sage: E = EllipticCurve([a/4, 1]) sage: E._reduce_model() Traceback (most recent call last): ... TypeError: _reduce_model() requires an integral model. """
# N.B. Must define s, r, t in the right order. else:
def _scale_by_units(self): r""" Return a model reduced with respect to scaling by units.
OUTPUT:
A model for this elliptic curve, optimally scaled with respect to scaling by units, with respect to the logarithmic embedding of |c4|^(1/4)+|c6|^(1/6). No scaling by roots of unity is carried out, so there is no change when the unit rank is 0.
EXAMPLES::
sage: K.<a> = NumberField(x^2-10) sage: u = K.units()[0] sage: E = EllipticCurve([0, 0, 0, 4536*a + 14148, -163728*a - 474336]) sage: E1 = E.scale_curve(u^5) sage: E1.ainvs() (0, 0, 0, 28087920796764302856*a + 88821804456186580548, -77225139016967233228487820912*a - 244207331916752959911655344864) sage: E1._scale_by_units().ainvs() (0, 0, 0, 4536*a + 14148, -163728*a - 474336)
A totally real cubic example::
sage: K.<a> = NumberField(x^3-x^2-6*x+5) sage: E = EllipticCurve([a + 1, a^2 + a - 1, a + 1, 44*a^2 + a - 258, -215*a^2 + 53*a + 1340]) sage: u1, u2 = K.units() sage: u = u1^2/u2^3 sage: E1 = E.scale_curve(u) sage: E1._scale_by_units().ainvs() == E.ainvs() True
A complex quartic example::
sage: K.<a> = CyclotomicField(5) sage: E = EllipticCurve([a + 1, a^2 + a - 1, a + 1, 44*a^2 + a - 258, -215*a^2 + 53*a + 1340]) sage: u = K.units()[0] sage: E1 = E.scale_curve(u^5) sage: E1._scale_by_units().ainvs() == E.ainvs() True """
def local_data(self, P=None, proof=None, algorithm="pari", globally=False): r""" Local data for this elliptic curve at the prime `P`.
INPUT:
- ``P`` -- either None, a prime ideal of the base field of self, or an element of the base field that generates a prime ideal.
- ``proof`` -- whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.
- ``algorithm`` (string, default: "pari") -- Ignored unless the base field is `\QQ`. If "pari", use the PARI C-library ``ellglobalred`` implementation of Tate's algorithm over `\QQ`. If "generic", use the general number field implementation.
- ``globally`` -- whether the local algorithm uses global generators for the prime ideals. Default is False, which won't require any information about the class group. If True, a generator for `P` will be used if `P` is principal. Otherwise, or if ``globally`` is False, the minimal model returned will preserve integrality at other primes, but not minimality.
OUTPUT:
If `P` is specified, returns the ``EllipticCurveLocalData`` object associated to the prime `P` for this curve. Otherwise, returns a list of such objects, one for each prime `P` in the support of the discriminant of this model.
.. note::
The model is not required to be integral on input.
EXAMPLES::
sage: K.<i> = NumberField(x^2+1) sage: E = EllipticCurve([1 + i, 0, 1, 0, 0]) sage: E.local_data() [Local data at Fractional ideal (2*i + 1): Reduction type: bad non-split multiplicative Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3 over Number Field in i with defining polynomial x^2 + 1 Minimal discriminant valuation: 1 Conductor exponent: 1 Kodaira Symbol: I1 Tamagawa Number: 1, Local data at Fractional ideal (-3*i - 2): Reduction type: bad split multiplicative Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3 over Number Field in i with defining polynomial x^2 + 1 Minimal discriminant valuation: 2 Conductor exponent: 1 Kodaira Symbol: I2 Tamagawa Number: 2] sage: E.local_data(K.ideal(3)) Local data at Fractional ideal (3): Reduction type: good Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3 over Number Field in i with defining polynomial x^2 + 1 Minimal discriminant valuation: 0 Conductor exponent: 0 Kodaira Symbol: I0 Tamagawa Number: 1 sage: E.local_data(2*i + 1) Local data at Fractional ideal (2*i + 1): Reduction type: bad non-split multiplicative Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3 over Number Field in i with defining polynomial x^2 + 1 Minimal discriminant valuation: 1 Conductor exponent: 1 Kodaira Symbol: I1 Tamagawa Number: 1
An example raised in :trac:`3897`::
sage: E = EllipticCurve([1,1]) sage: E.local_data(3) Local data at Principal ideal (3) of Integer Ring: Reduction type: good Local minimal model: Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field Minimal discriminant valuation: 0 Conductor exponent: 0 Kodaira Symbol: I0 Tamagawa Number: 1
""" # We use the "number_field" flag because the actual proof dependence is in PARI's number field functions.
def _get_local_data(self, P, proof, algorithm="pari", globally=False): r""" Internal function to create data for this elliptic curve at the prime `P`.
This function handles the caching of local data. It is called by local_data() which is the user interface and which parses the input parameters `P` and proof.
INPUT:
- ``P`` -- either None or a prime ideal of the base field of self.
- ``proof`` -- whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.
- ``algorithm`` (string, default: "pari") -- Ignored unless the base field is `\QQ`. If "pari", use the PARI C-library ``ellglobalred`` implementation of Tate's algorithm over `\QQ`. If "generic", use the general number field implementation.
- ``globally`` -- whether the local algorithm uses global generators for the prime ideals. Default is False, which won't require any information about the class group. If True, a generator for `P` will be used if `P` is principal. Otherwise, or if ``globally`` is False, the minimal model returned will preserve integrality at other primes, but not minimality.
EXAMPLES::
sage: K.<i> = NumberField(x^2+1) sage: E = EllipticCurve(K,[0,1,0,-160,308]) sage: p = K.ideal(i+1) sage: E._get_local_data(p, False) Local data at Fractional ideal (i + 1): Reduction type: good Local minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + (-10)*x + (-10) over Number Field in i with defining polynomial x^2 + 1 Minimal discriminant valuation: 0 Conductor exponent: 0 Kodaira Symbol: I0 Tamagawa Number: 1
Verify that we cache based on the proof value and the algorithm choice::
sage: E._get_local_data(p, False) is E._get_local_data(p, True) False
sage: E._get_local_data(p, None, "pari") is E._get_local_data(p, None, "generic") False """
def local_minimal_model(self, P, proof = None, algorithm="pari"): r""" Returns a model which is integral at all primes and minimal at `P`.
INPUT:
- ``P`` -- either None or a prime ideal of the base field of self.
- ``proof`` -- whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.
- ``algorithm`` (string, default: "pari") -- Ignored unless the base field is `\QQ`. If "pari", use the PARI C-library ``ellglobalred`` implementation of Tate's algorithm over `\QQ`. If "generic", use the general number field implementation.
OUTPUT:
A model of the curve which is minimal (and integral) at `P`.
.. note::
The model is not required to be integral on input.
For principal `P`, a generator is used as a uniformizer, and integrality or minimality at other primes is not affected. For non-principal `P`, the minimal model returned will preserve integrality at other primes, but not minimality.
EXAMPLES::
sage: K.<a>=NumberField(x^2-5) sage: E=EllipticCurve([20, 225, 750, 1250*a + 6250, 62500*a + 15625]) sage: P=K.ideal(a) sage: E.local_minimal_model(P).ainvs() (0, 1, 0, 2*a - 34, -4*a + 66) """ # We use the "number_field" flag because the actual proof dependence is in PARI's number field functions.
def has_good_reduction(self, P): r""" Return True if this elliptic curve has good reduction at the prime `P`.
INPUT:
- ``P`` -- a prime ideal of the base field of self, or a field element generating such an ideal.
OUTPUT:
(bool) -- True if the curve has good reduction at `P`, else False.
.. note::
This requires determining a local integral minimal model; we do not just check that the discriminant of the current model has valuation zero.
EXAMPLES::
sage: E=EllipticCurve('14a1') sage: [(p,E.has_good_reduction(p)) for p in prime_range(15)] [(2, False), (3, True), (5, True), (7, False), (11, True), (13, True)]
sage: K.<a>=NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.has_good_reduction(p)) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), True), (Fractional ideal (2*a + 1), False)] """
def has_bad_reduction(self, P): r""" Return True if this elliptic curve has bad reduction at the prime `P`.
INPUT:
- ``P`` -- a prime ideal of the base field of self, or a field element generating such an ideal.
OUTPUT:
(bool) True if the curve has bad reduction at `P`, else False.
.. note::
This requires determining a local integral minimal model; we do not just check that the discriminant of the current model has valuation zero.
EXAMPLES::
sage: E=EllipticCurve('14a1') sage: [(p,E.has_bad_reduction(p)) for p in prime_range(15)] [(2, True), (3, False), (5, False), (7, True), (11, False), (13, False)]
sage: K.<a>=NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.has_bad_reduction(p)) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), True)] """
def has_multiplicative_reduction(self, P): r""" Return True if this elliptic curve has (bad) multiplicative reduction at the prime `P`.
.. note::
See also ``has_split_multiplicative_reduction()`` and ``has_nonsplit_multiplicative_reduction()``.
INPUT:
- ``P`` -- a prime ideal of the base field of self, or a field element generating such an ideal.
OUTPUT:
(bool) True if the curve has multiplicative reduction at `P`, else False.
EXAMPLES::
sage: E=EllipticCurve('14a1') sage: [(p,E.has_multiplicative_reduction(p)) for p in prime_range(15)] [(2, True), (3, False), (5, False), (7, True), (11, False), (13, False)]
sage: K.<a>=NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.has_multiplicative_reduction(p)) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), False)] """
def has_split_multiplicative_reduction(self, P): r""" Return True if this elliptic curve has (bad) split multiplicative reduction at the prime `P`.
INPUT:
- ``P`` -- a prime ideal of the base field of self, or a field element generating such an ideal.
OUTPUT:
(bool) True if the curve has split multiplicative reduction at `P`, else False.
EXAMPLES::
sage: E=EllipticCurve('14a1') sage: [(p,E.has_split_multiplicative_reduction(p)) for p in prime_range(15)] [(2, False), (3, False), (5, False), (7, True), (11, False), (13, False)]
sage: K.<a>=NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.has_split_multiplicative_reduction(p)) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), False)] """
def has_nonsplit_multiplicative_reduction(self, P): r""" Return True if this elliptic curve has (bad) non-split multiplicative reduction at the prime `P`.
INPUT:
- ``P`` -- a prime ideal of the base field of self, or a field element generating such an ideal.
OUTPUT:
(bool) True if the curve has non-split multiplicative reduction at `P`, else False.
EXAMPLES::
sage: E=EllipticCurve('14a1') sage: [(p,E.has_nonsplit_multiplicative_reduction(p)) for p in prime_range(15)] [(2, True), (3, False), (5, False), (7, False), (11, False), (13, False)]
sage: K.<a>=NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.has_nonsplit_multiplicative_reduction(p)) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), False)] """
def has_additive_reduction(self, P): r""" Return True if this elliptic curve has (bad) additive reduction at the prime `P`.
INPUT:
- ``P`` -- a prime ideal of the base field of self, or a field element generating such an ideal.
OUTPUT:
(bool) True if the curve has additive reduction at `P`, else False.
EXAMPLES::
sage: E=EllipticCurve('27a1') sage: [(p,E.has_additive_reduction(p)) for p in prime_range(15)] [(2, False), (3, True), (5, False), (7, False), (11, False), (13, False)]
sage: K.<a>=NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.has_additive_reduction(p)) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), True)] """
def tamagawa_number(self, P, proof = None): r""" Returns the Tamagawa number of this elliptic curve at the prime `P`.
INPUT:
- ``P`` -- either None or a prime ideal of the base field of self.
- ``proof`` -- whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.
OUTPUT:
(positive integer) The Tamagawa number of the curve at `P`.
EXAMPLES::
sage: K.<a>=NumberField(x^2-5) sage: E=EllipticCurve([20, 225, 750, 625*a + 6875, 31250*a + 46875]) sage: [E.tamagawa_number(P) for P in E.discriminant().support()] [1, 1, 1, 1] sage: K.<a> = QuadraticField(-11) sage: E = EllipticCurve('11a1').change_ring(K) sage: [E.tamagawa_number(P) for P in K(11).support()] [10] """ # We use the "number_field" flag because the actual proof dependence is in PARI's number field functions.
def tamagawa_numbers(self): """ Return a list of all Tamagawa numbers for all prime divisors of the conductor (in order).
EXAMPLES::
sage: e = EllipticCurve('30a1') sage: e.tamagawa_numbers() [2, 3, 1] sage: vector(e.tamagawa_numbers()) (2, 3, 1) sage: K.<a>=NumberField(x^2+3) sage: eK = e.base_extend(K) sage: eK.tamagawa_numbers() [4, 6, 1] """
def tamagawa_exponent(self, P, proof = None): r""" Returns the Tamagawa index of this elliptic curve at the prime `P`.
INPUT:
- ``P`` -- either None or a prime ideal of the base field of self.
- ``proof`` -- whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.
OUTPUT:
(positive integer) The Tamagawa index of the curve at P.
EXAMPLES::
sage: K.<a>=NumberField(x^2-5) sage: E=EllipticCurve([20, 225, 750, 625*a + 6875, 31250*a + 46875]) sage: [E.tamagawa_exponent(P) for P in E.discriminant().support()] [1, 1, 1, 1] sage: K.<a> = QuadraticField(-11) sage: E = EllipticCurve('11a1').change_ring(K) sage: [E.tamagawa_exponent(P) for P in K(11).support()] [10] """ # We use the "number_field" flag because the actual proof dependence is in PARI's number field functions.
def tamagawa_product_bsd(self): r""" Given an elliptic curve `E` over a number field `K`, this function returns the integer `C(E/K)` that appears in the Birch and Swinnerton-Dyer conjecture accounting for the local information at finite places. If the model is a global minimal model then `C(E/K)` is simply the product of the Tamagawa numbers `c_v` where `v` runs over all prime ideals of `K`. Otherwise, if the model has to be changed at a place `v` a correction factor appears. The definition is such that `C(E/K)` times the periods at the infinite places is invariant under change of the Weierstrass model. See [Tate1966]_ and [DD2010]_ for details.
.. note::
This definition is slightly different from the definition of ``tamagawa_product`` for curves defined over `\QQ`. Over the rational number it is always defined to be the product of the Tamagawa numbers, so the two definitions only agree when the model is global minimal.
OUTPUT:
A rational number
EXAMPLES::
sage: K.<i> = NumberField(x^2+1) sage: E = EllipticCurve([0,2+i]) sage: E.tamagawa_product_bsd() 1
sage: E = EllipticCurve([(2*i+1)^2,i*(2*i+1)^7]) sage: E.tamagawa_product_bsd() 4
An example where the Neron model changes over K::
sage: K.<t> = NumberField(x^5-10*x^3+5*x^2+10*x+1) sage: E = EllipticCurve(K,'75a1') sage: E.tamagawa_product_bsd() 5 sage: da = E.local_data() sage: [dav.tamagawa_number() for dav in da] [1, 1]
An example over `\QQ` (:trac:`9413`)::
sage: E = EllipticCurve('30a') sage: E.tamagawa_product_bsd() 6 """ # uu is the quotient of the Neron differential at pp divided by # the differential associated to this particular equation E else:
def kodaira_symbol(self, P, proof = None): r""" Returns the Kodaira Symbol of this elliptic curve at the prime `P`.
INPUT:
- ``P`` -- either None or a prime ideal of the base field of self.
- ``proof`` -- whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.
OUTPUT:
The Kodaira Symbol of the curve at P, represented as a string.
EXAMPLES::
sage: K.<a>=NumberField(x^2-5) sage: E=EllipticCurve([20, 225, 750, 625*a + 6875, 31250*a + 46875]) sage: bad_primes = E.discriminant().support(); bad_primes [Fractional ideal (-a), Fractional ideal (7/2*a - 81/2), Fractional ideal (-a - 52), Fractional ideal (2)] sage: [E.kodaira_symbol(P) for P in bad_primes] [I0, I1, I1, II] sage: K.<a> = QuadraticField(-11) sage: E = EllipticCurve('11a1').change_ring(K) sage: [E.kodaira_symbol(P) for P in K(11).support()] [I10] """ # We use the "number_field" flag because the actual proof dependence is in PARI's number field functions.
def conductor(self): r""" Returns the conductor of this elliptic curve as a fractional ideal of the base field.
OUTPUT:
(fractional ideal) The conductor of the curve.
EXAMPLES::
sage: K.<i>=NumberField(x^2+1) sage: EllipticCurve([i, i - 1, i + 1, 24*i + 15, 14*i + 35]).conductor() Fractional ideal (21*i - 3) sage: K.<a>=NumberField(x^2-x+3) sage: EllipticCurve([1 + a , -1 + a , 1 + a , -11 + a , 5 -9*a ]).conductor() Fractional ideal (-6*a)
A not so well known curve with everywhere good reduction::
sage: K.<a>=NumberField(x^2-38) sage: E=EllipticCurve([0,0,0, 21796814856932765568243810*a - 134364590724198567128296995, 121774567239345229314269094644186997594*a - 750668847495706904791115375024037711300]) sage: E.conductor() Fractional ideal (1)
An example which used to fail (see :trac:`5307`)::
sage: K.<w>=NumberField(x^2+x+6) sage: E=EllipticCurve([w,-1,0,-w-6,0]) sage: E.conductor() Fractional ideal (86304, w + 5898)
An example raised in :trac:`11346`::
sage: K.<g> = NumberField(x^2 - x - 1) sage: E1 = EllipticCurve(K,[0,0,0,-1/48,-161/864]) sage: [(p.smallest_integer(),e) for p,e in E1.conductor().factor()] [(2, 4), (3, 1), (5, 1)] """
# Note: for number fields other than QQ we could initialize # N=K.ideal(1) or N=OK.ideal(1), which are the same, but for # K == QQ it has to be ZZ.ideal(1). for d in self.local_data()],\ OK.ideal(1))
def minimal_discriminant_ideal(self): r""" Return the minimal discriminant ideal of this elliptic curve.
OUTPUT:
The integral ideal `D` whose valuation at every prime `P` is that of the local minimal model for `E` at `P`. If `E` has a global minimal model, this will be the principal ideal generated by the discriminant of any such model, but otherwise it can be a proper divisor of the discriminant of any model.
EXAMPLES::
sage: K.<a> = NumberField(x^2-x-57) sage: K.class_number() 3 sage: E = EllipticCurve([a,-a,a,-5692-820*a,-259213-36720*a]) sage: K.ideal(E.discriminant()) Fractional ideal (90118662980*a + 636812084644) sage: K.ideal(E.discriminant()).factor() (Fractional ideal (2))^2 * (Fractional ideal (3, a + 2))^12
Here the minimal discriminant ideal is principal but there is no global minimal model since the quotient is the 12th power of a non-principal ideal::
sage: E.minimal_discriminant_ideal() Fractional ideal (4) sage: E.minimal_discriminant_ideal().factor() (Fractional ideal (2))^2
If (and only if) the curve has everywhere good reduction the result is the unit ideal::
sage: K.<a> = NumberField(x^2-26) sage: E = EllipticCurve([a,a-1,a+1,4*a+10,2*a+6]) sage: E.conductor() Fractional ideal (1) sage: E.discriminant() -104030*a - 530451 sage: E.minimal_discriminant_ideal() Fractional ideal (1)
Over `\QQ`, the result returned is an ideal of `\ZZ` rather than a fractional ideal of `\QQ`::
sage: E = EllipticCurve([1,2,3,4,5]) sage: E.minimal_discriminant_ideal() Principal ideal (10351) of Integer Ring """ # we treat separately the case where there are no bad primes, # which cannot happen over QQ, since ideals of QQ behave # differently to (fractional) ideals of other number fields.
def non_minimal_primes(self): r""" Returns a list of primes at which this elliptic curve is not minimal.
OUTPUT:
A list of prime ideals (or prime numbers when the base field is `\QQ`, empty if this is a global minimal model.
EXAMPLES::
sage: K.<a> = NumberField(x^2-10) sage: E = EllipticCurve([0, 0, 0, -22500, 750000*a]) sage: E.non_minimal_primes() [Fractional ideal (2, a), Fractional ideal (5, a)] sage: K.ideal(E.discriminant()).factor() (Fractional ideal (2, a))^24 * (Fractional ideal (3, a + 1))^5 * (Fractional ideal (3, a + 2))^5 * (Fractional ideal (5, a))^24 * (Fractional ideal (7)) sage: E.minimal_discriminant_ideal().factor() (Fractional ideal (2, a))^12 * (Fractional ideal (3, a + 1))^5 * (Fractional ideal (3, a + 2))^5 * (Fractional ideal (7))
Over `\QQ`, the primes returned are integers, not ideals::
sage: E = EllipticCurve([0,0,0,-3024,46224]) sage: E.non_minimal_primes() [2, 3] sage: Emin = E.global_minimal_model() sage: Emin.non_minimal_primes() []
If the model is not globally integral, a ``ValueError`` is raised::
sage: E = EllipticCurve([0,0,0,1/2,1/3]) sage: E.non_minimal_primes() Traceback (most recent call last): ... ValueError: non_minimal_primes only defined for integral models """
def is_global_minimal_model(self): r""" Returns whether this elliptic curve is a global minimal model.
OUTPUT:
Boolean, False if E is not integral, or if E is non-minimal at some prime, else True.
EXAMPLES::
sage: K.<a> = NumberField(x^2-10) sage: E = EllipticCurve([0, 0, 0, -22500, 750000*a]) sage: E.is_global_minimal_model() False sage: E.non_minimal_primes() [Fractional ideal (2, a), Fractional ideal (5, a)]
sage: E = EllipticCurve([0,0,0,-3024,46224]) sage: E.is_global_minimal_model() False sage: E.non_minimal_primes() [2, 3] sage: Emin = E.global_minimal_model() sage: Emin.is_global_minimal_model() True
A necessary condition to be a global minimal model is that the model must be globally integral::
sage: E = EllipticCurve([0,0,0,1/2,1/3]) sage: E.is_global_minimal_model() False sage: Emin.is_global_minimal_model() True sage: Emin.ainvs() (0, 1, 1, -2, 0) """
def global_minimality_class(self): r""" Returns the obstruction to this curve having a global minimal model.
OUTPUT:
An ideal class of the base number field, which is trivial if and only if the elliptic curve has a global minimal model, and which can be used to find global and semi-global minimal models.
EXAMPLES:
A curve defined over a field of class number 2 with no global minimal model was a nontrivial minimality class::
sage: K.<a> = NumberField(x^2-10) sage: K.class_number() 2 sage: E = EllipticCurve([0, 0, 0, -22500, 750000*a]) sage: E.global_minimality_class() Fractional ideal class (10, 5*a) sage: E.global_minimality_class().order() 2
Over the same field, a curve defined by a non-minimal model has trivial class, showing that a global minimal model does exist::
sage: K.<a> = NumberField(x^2-10) sage: E = EllipticCurve([0,0,0,4536*a+14148,-163728*a- 474336]) sage: E.is_global_minimal_model() False sage: E.global_minimality_class() Trivial principal fractional ideal class
Over a field of class number 1 the result is always the trivial class::
sage: K.<a> = NumberField(x^2-5) sage: E = EllipticCurve([0, 0, 0, K(16), K(64)]) sage: E.global_minimality_class() Trivial principal fractional ideal class
sage: E = EllipticCurve([0, 0, 0, 16, 64]) sage: E.base_field() Rational Field sage: E.global_minimality_class() 1 """ K.ideal(1))
def has_global_minimal_model(self): r""" Returns whether this elliptic curve has a global minimal model.
OUTPUT:
Boolean, True iff a global minimal model exists, i.e. an integral model which is minimal at every prime.
EXAMPLES::
sage: K.<a> = NumberField(x^2-10) sage: E = EllipticCurve([0,0,0,4536*a+14148,-163728*a-474336]) sage: E.is_global_minimal_model() False sage: E.has_global_minimal_model() True """
def global_minimal_model(self, proof = None, semi_global=False): r""" Returns a model of self that is integral, and minimal.
.. note::
Over fields of class number greater than 1, a global minimal model may not exist. If it does not, set the parameter ``semi_global`` to ``True`` to obtain a model minimal at all but one prime.
INPUT:
- ``proof`` -- whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.
- ``semi_global`` (boolean, default False) -- if there is no global minimal mode, return a semi-global minimal model (minimal at all but one prime) instead, if True; raise an error if False. No effect if a global minimal model exists.
OUTPUT:
A global integral and minimal model, or an integral model minimal at all but one prime of there is no global minimal model and the flag ``semi_global`` is True.
EXAMPLES::
sage: K.<a> = NumberField(x^2-38) sage: E = EllipticCurve([0,0,0, 21796814856932765568243810*a - 134364590724198567128296995, 121774567239345229314269094644186997594*a - 750668847495706904791115375024037711300]) sage: E2 = E.global_minimal_model() sage: E2 Elliptic Curve defined by y^2 + a*x*y + (a+1)*y = x^3 + (a+1)*x^2 + (4*a+15)*x + (4*a+21) over Number Field in a with defining polynomial x^2 - 38 sage: E2.local_data() []
See :trac:`11347`::
sage: K.<g> = NumberField(x^2 - x - 1) sage: E = EllipticCurve(K,[0,0,0,-1/48,161/864]).integral_model().global_minimal_model(); E Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 over Number Field in g with defining polynomial x^2 - x - 1 sage: [(p.norm(), e) for p, e in E.conductor().factor()] [(9, 1), (5, 1)] sage: [(p.norm(), e) for p, e in E.discriminant().factor()] [(-5, 2), (9, 1)]
See :trac:`14472`, this used not to work over a relative extension::
sage: K1.<w> = NumberField(x^2+x+1) sage: m = polygen(K1) sage: K2.<v> = K1.extension(m^2-w+1) sage: E = EllipticCurve([0*v,-432]) sage: E.global_minimal_model() Elliptic Curve defined by y^2 + y = x^3 over Number Field in v with defining polynomial x^2 - w + 1 over its base field
See :trac:`18662`: for fields of class number greater than 1, even when global minimal models did exist, their computation was not implemented. Now it is::
sage: K.<a> = NumberField(x^2-10) sage: K.class_number() 2 sage: E = EllipticCurve([0,0,0,-186408*a - 589491, 78055704*a + 246833838]) sage: E.discriminant().norm() 16375845905239507992576 sage: E.discriminant().norm().factor() 2^31 * 3^27 sage: E.has_global_minimal_model() True sage: Emin = E.global_minimal_model(); Emin Elliptic Curve defined by y^2 + (a+1)*x*y + (a+1)*y = x^3 + (-a)*x^2 + (a-12)*x + (-2*a+2) over Number Field in a with defining polynomial x^2 - 10 sage: Emin.discriminant().norm() 3456 sage: Emin.discriminant().norm().factor() 2^7 * 3^3
If there is no global minimal model, this method will raise an error unless you set the parameter ``semi_global`` to ``True``::
sage: K.<a> = NumberField(x^2-10) sage: K.class_number() 2 sage: E = EllipticCurve([a,a,0,3*a+8,4*a+3]) sage: E.has_global_minimal_model() False sage: E.global_minimal_model() Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 + a*x*y = x^3 + a*x^2 + (3*a+8)*x + (4*a+3) over Number Field in a with defining polynomial x^2 - 10 has no global minimal model! For a semi-global minimal model use semi_global=True sage: E.global_minimal_model(semi_global=True) Elliptic Curve defined by y^2 + a*x*y = x^3 + a*x^2 + (3*a+8)*x + (4*a+3) over Number Field in a with defining polynomial x^2 - 10
An example of a curve with everywhere good reduction but which has no model with unit discriminant::
sage: K.<a> = NumberField(x^2-x-16) sage: K.class_number() 2 sage: E = EllipticCurve([0,0,0,-15221331*a - 53748576, -79617688290*a - 281140318368]) sage: Emin = E.global_minimal_model(semi_global=True) sage: Emin.ainvs() (a, a - 1, a, 605*a - 2728, 15887*a - 71972) sage: Emin.discriminant() -17*a - 16 sage: Emin.discriminant().norm() -4096 sage: Emin.minimal_discriminant_ideal() Fractional ideal (1) sage: E.conductor() Fractional ideal (1) """ # We use the "number_field" flag because the actual proof dependence is in PARI's number field functions.
else:
def reduction(self,place): r""" Return the reduction of the elliptic curve at a place of good reduction.
INPUT:
- ``place`` -- a prime ideal in the base field of the curve
OUTPUT:
An elliptic curve over a finite field, the residue field of the place.
EXAMPLES::
sage: K.<i> = QuadraticField(-1) sage: EK = EllipticCurve([0,0,0,i,i+3]) sage: v = K.fractional_ideal(2*i+3) sage: EK.reduction(v) Elliptic Curve defined by y^2 = x^3 + 5*x + 8 over Residue field of Fractional ideal (2*i + 3) sage: EK.reduction(K.ideal(1+i)) Traceback (most recent call last): ... ValueError: The curve must have good reduction at the place. sage: EK.reduction(K.ideal(2)) Traceback (most recent call last): ... ValueError: The ideal must be prime. sage: K=QQ.extension(x^2+x+1,"a") sage: E=EllipticCurve([1024*K.0,1024*K.0]) sage: E.reduction(2*K) Elliptic Curve defined by y^2 + (abar+1)*y = x^3 over Residue field in abar of Fractional ideal (2) """ except TypeError: raise TypeError("The parameter must be an ideal of the base field of the elliptic curve")
def _torsion_bound(self,number_of_places = 20): r""" An upper bound on the order of the torsion subgroup.
INPUT:
- ``number_of_places`` (positive integer, default = 20) -- the number of places that will be used to find the bound.
OUTPUT:
(integer) An upper bound on the torsion order.
ALGORITHM:
An upper bound on the order of the torsion.group of the elliptic curve is obtained by counting points modulo several primes of good reduction. Note that the upper bound returned by this function is a multiple of the order of the torsion group, and in general will be greater than the order.
EXAMPLES::
sage: CDB=CremonaDatabase() sage: [E._torsion_bound() for E in CDB.iter([14])] [6, 6, 6, 6, 6, 6] sage: [E.torsion_order() for E in CDB.iter([14])] [6, 6, 2, 6, 2, 6]
An example over a relative number field (see :trac:`16011`)::
sage: R.<x> = QQ[] sage: F.<a> = QuadraticField(5) sage: K.<b> = F.extension(x^2-3) sage: E = EllipticCurve(K,[0,0,0,b,1]) sage: E.torsion_subgroup().order() 1
""" # runs through primes, decomposes them into prime ideals # runs through prime ideals above p # take only places with small residue field (so that the # number of points will be small) # check if the model is integral at the place # check if the formal group at the place is torsion-free # if so the torsion injects into the reduction
@cached_method def torsion_subgroup(self): r""" Returns the torsion subgroup of this elliptic curve.
OUTPUT:
(``EllipticCurveTorsionSubgroup``) The ``EllipticCurveTorsionSubgroup`` associated to this elliptic curve.
EXAMPLES::
sage: E = EllipticCurve('11a1') sage: K.<t>=NumberField(x^4 + x^3 + 11*x^2 + 41*x + 101) sage: EK = E.base_extend(K) sage: tor = EK.torsion_subgroup() # long time (2s on sage.math, 2014) sage: tor # long time Torsion Subgroup isomorphic to Z/5 + Z/5 associated to the Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in t with defining polynomial x^4 + x^3 + 11*x^2 + 41*x + 101 sage: tor.gens() # long time ((16 : 60 : 1), (t : 1/11*t^3 + 6/11*t^2 + 19/11*t + 48/11 : 1))
::
sage: E = EllipticCurve('15a1') sage: K.<t>=NumberField(x^2 + 2*x + 10) sage: EK=E.base_extend(K) sage: EK.torsion_subgroup() Torsion Subgroup isomorphic to Z/4 + Z/4 associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + (-10)*x + (-10) over Number Field in t with defining polynomial x^2 + 2*x + 10
::
sage: E = EllipticCurve('19a1') sage: K.<t>=NumberField(x^9-3*x^8-4*x^7+16*x^6-3*x^5-21*x^4+5*x^3+7*x^2-7*x+1) sage: EK=E.base_extend(K) sage: EK.torsion_subgroup() Torsion Subgroup isomorphic to Z/9 associated to the Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-9)*x + (-15) over Number Field in t with defining polynomial x^9 - 3*x^8 - 4*x^7 + 16*x^6 - 3*x^5 - 21*x^4 + 5*x^3 + 7*x^2 - 7*x + 1
::
sage: K.<i> = QuadraticField(-1) sage: EK = EllipticCurve([0,0,0,i,i+3]) sage: EK.torsion_subgroup () Torsion Subgroup isomorphic to Trivial group associated to the Elliptic Curve defined by y^2 = x^3 + i*x + (i+3) over Number Field in i with defining polynomial x^2 + 1 """
@cached_method def torsion_order(self): r""" Returns the order of the torsion subgroup of this elliptic curve.
OUTPUT:
(integer) the order of the torsion subgroup of this elliptic curve.
EXAMPLES::
sage: E = EllipticCurve('11a1') sage: K.<t> = NumberField(x^4 + x^3 + 11*x^2 + 41*x + 101) sage: EK = E.base_extend(K) sage: EK.torsion_order() # long time (2s on sage.math, 2014) 25
::
sage: E = EllipticCurve('15a1') sage: K.<t> = NumberField(x^2 + 2*x + 10) sage: EK = E.base_extend(K) sage: EK.torsion_order() 16
::
sage: E = EllipticCurve('19a1') sage: K.<t> = NumberField(x^9-3*x^8-4*x^7+16*x^6-3*x^5-21*x^4+5*x^3+7*x^2-7*x+1) sage: EK = E.base_extend(K) sage: EK.torsion_order() 9
::
sage: K.<i> = QuadraticField(-1) sage: EK = EllipticCurve([0,0,0,i,i+3]) sage: EK.torsion_order() 1 """
def torsion_points(self): r""" Returns a list of the torsion points of this elliptic curve.
OUTPUT:
(list) A sorted list of the torsion points.
EXAMPLES::
sage: E = EllipticCurve('11a1') sage: E.torsion_points() [(0 : 1 : 0), (5 : -6 : 1), (5 : 5 : 1), (16 : -61 : 1), (16 : 60 : 1)] sage: K.<t> = NumberField(x^4 + x^3 + 11*x^2 + 41*x + 101) sage: EK = E.base_extend(K) sage: EK.torsion_points() # long time (1s on sage.math, 2014) [(0 : 1 : 0), (16 : 60 : 1), (5 : 5 : 1), (5 : -6 : 1), (16 : -61 : 1), (t : 1/11*t^3 + 6/11*t^2 + 19/11*t + 48/11 : 1), (-3/55*t^3 - 7/55*t^2 - 2/55*t - 133/55 : 6/55*t^3 + 3/55*t^2 + 25/11*t + 156/55 : 1), (-9/121*t^3 - 21/121*t^2 - 127/121*t - 377/121 : -7/121*t^3 + 24/121*t^2 + 197/121*t + 16/121 : 1), (5/121*t^3 - 14/121*t^2 - 158/121*t - 453/121 : -49/121*t^3 - 129/121*t^2 - 315/121*t - 207/121 : 1), (10/121*t^3 + 49/121*t^2 + 168/121*t + 73/121 : 32/121*t^3 + 60/121*t^2 - 261/121*t - 807/121 : 1), (1/11*t^3 - 5/11*t^2 + 19/11*t - 40/11 : -6/11*t^3 - 3/11*t^2 - 26/11*t - 321/11 : 1), (14/121*t^3 - 15/121*t^2 + 90/121*t + 232/121 : 16/121*t^3 - 69/121*t^2 + 293/121*t - 46/121 : 1), (3/55*t^3 + 7/55*t^2 + 2/55*t + 78/55 : 7/55*t^3 - 24/55*t^2 + 9/11*t + 17/55 : 1), (-5/121*t^3 + 36/121*t^2 - 84/121*t + 24/121 : 34/121*t^3 - 27/121*t^2 + 305/121*t + 708/121 : 1), (-26/121*t^3 + 20/121*t^2 - 219/121*t - 995/121 : 15/121*t^3 + 156/121*t^2 - 232/121*t + 2766/121 : 1), (1/11*t^3 - 5/11*t^2 + 19/11*t - 40/11 : 6/11*t^3 + 3/11*t^2 + 26/11*t + 310/11 : 1), (-26/121*t^3 + 20/121*t^2 - 219/121*t - 995/121 : -15/121*t^3 - 156/121*t^2 + 232/121*t - 2887/121 : 1), (-5/121*t^3 + 36/121*t^2 - 84/121*t + 24/121 : -34/121*t^3 + 27/121*t^2 - 305/121*t - 829/121 : 1), (3/55*t^3 + 7/55*t^2 + 2/55*t + 78/55 : -7/55*t^3 + 24/55*t^2 - 9/11*t - 72/55 : 1), (14/121*t^3 - 15/121*t^2 + 90/121*t + 232/121 : -16/121*t^3 + 69/121*t^2 - 293/121*t - 75/121 : 1), (t : -1/11*t^3 - 6/11*t^2 - 19/11*t - 59/11 : 1), (10/121*t^3 + 49/121*t^2 + 168/121*t + 73/121 : -32/121*t^3 - 60/121*t^2 + 261/121*t + 686/121 : 1), (5/121*t^3 - 14/121*t^2 - 158/121*t - 453/121 : 49/121*t^3 + 129/121*t^2 + 315/121*t + 86/121 : 1), (-9/121*t^3 - 21/121*t^2 - 127/121*t - 377/121 : 7/121*t^3 - 24/121*t^2 - 197/121*t - 137/121 : 1), (-3/55*t^3 - 7/55*t^2 - 2/55*t - 133/55 : -6/55*t^3 - 3/55*t^2 - 25/11*t - 211/55 : 1)]
::
sage: E = EllipticCurve('15a1') sage: K.<t> = NumberField(x^2 + 2*x + 10) sage: EK = E.base_extend(K) sage: EK.torsion_points() [(-7 : -5*t - 2 : 1), (-7 : 5*t + 8 : 1), (-13/4 : 9/8 : 1), (-2 : -2 : 1), (-2 : 3 : 1), (-t - 2 : -t - 7 : 1), (-t - 2 : 2*t + 8 : 1), (-1 : 0 : 1), (t : t - 5 : 1), (t : -2*t + 4 : 1), (0 : 1 : 0), (1/2 : -5/4*t - 2 : 1), (1/2 : 5/4*t + 1/2 : 1), (3 : -2 : 1), (8 : -27 : 1), (8 : 18 : 1)]
::
sage: K.<i> = QuadraticField(-1) sage: EK = EllipticCurve(K,[0,0,0,0,-1]) sage: EK.torsion_points () [(-2 : -3*i : 1), (-2 : 3*i : 1), (0 : -i : 1), (0 : i : 1), (0 : 1 : 0), (1 : 0 : 1)] """
def rank_bounds(self, **kwds): r""" Returns the lower and upper bounds using :meth:`~simon_two_descent`. The results of :meth:`~simon_two_descent` are cached.
.. NOTE::
The optional parameters control the Simon two descent algorithm; see the documentation of :meth:`~simon_two_descent` for more details.
INPUT:
- ``verbose`` -- 0, 1, 2, or 3 (default: 0), the verbosity level
- ``lim1`` -- (default: 2) limit on trivial points on quartics
- ``lim3`` -- (default: 4) limit on points on ELS quartics
- ``limtriv`` -- (default: 2) limit on trivial points on elliptic curve
- ``maxprob`` -- (default: 20)
- ``limbigprime`` -- (default: 30) to distinguish between small and large prime numbers. Use probabilistic tests for large primes. If 0, don't use probabilistic tests.
- ``known_points`` -- (default: None) list of known points on the curve
OUTPUT:
lower and upper bounds for the rank of the Mordell-Weil group
.. NOTE::
For non-quadratic number fields, this code does return, but it takes a long time.
EXAMPLES::
sage: K.<a> = NumberField(x^2 + 23, 'a') sage: E = EllipticCurve(K, '37') sage: E == loads(dumps(E)) True sage: E.rank_bounds() (2, 2)
Here is a curve with two-torsion, again the bounds coincide::
sage: Qrt5.<rt5>=NumberField(x^2-5) sage: E=EllipticCurve([0,5-rt5,0,rt5,0]) sage: E.rank_bounds() (1, 1)
Finally an example with non-trivial 2-torsion in Sha. So the 2-descent will not be able to determine the rank, but can only give bounds::
sage: E = EllipticCurve("15a5") sage: K.<t> = NumberField(x^2-6) sage: EK = E.base_extend(K) sage: EK.rank_bounds(lim1=1,lim3=1,limtriv=1) (0, 2)
IMPLEMENTATION:
Uses Denis Simon's PARI/GP scripts from http://www.math.unicaen.fr/~simon/.
""" # this was corrected in trac 13593. upper is the dimension # of the 2-selmer group, so we can certainly remove the # 2-torsion of the Mordell-Weil group.
def rank(self, **kwds): r""" Return the rank of this elliptic curve, if it can be determined.
.. NOTE::
The optional parameters control the Simon two descent algorithm; see the documentation of :meth:`~simon_two_descent` for more details.
INPUT:
- ``verbose`` -- 0, 1, 2, or 3 (default: 0), the verbosity level
- ``lim1`` -- (default: 2) limit on trivial points on quartics
- ``lim3`` -- (default: 4) limit on points on ELS quartics
- ``limtriv`` -- (default: 2) limit on trivial points on elliptic curve
- ``maxprob`` -- (default: 20)
- ``limbigprime`` -- (default: 30) to distinguish between small and large prime numbers. Use probabilistic tests for large primes. If 0, don't use probabilistic tests.
- ``known_points`` -- (default: None) list of known points on the curve
OUTPUT:
If the upper and lower bounds given by Simon two-descent are the same, then the rank has been uniquely identified and we return this. Otherwise, we raise a ValueError with an error message specifying the upper and lower bounds.
.. NOTE::
For non-quadratic number fields, this code does return, but it takes a long time.
EXAMPLES::
sage: K.<a> = NumberField(x^2 + 23, 'a') sage: E = EllipticCurve(K, '37') sage: E == loads(dumps(E)) True sage: E.rank() 2
Here is a curve with two-torsion in the Tate-Shafarevich group, so here the bounds given by the algorithm do not uniquely determine the rank::
sage: E = EllipticCurve("15a5") sage: K.<t> = NumberField(x^2-6) sage: EK = E.base_extend(K) sage: EK.rank(lim1=1, lim3=1, limtriv=1) Traceback (most recent call last): ... ValueError: There is insufficient data to determine the rank - 2-descent gave lower bound 0 and upper bound 2
IMPLEMENTATION:
Uses Denis Simon's PARI/GP scripts from http://www.math.unicaen.fr/~simon/.
""" else:
def gens(self, **kwds): r""" Return some points of infinite order on this elliptic curve.
Contrary to what the name of this method suggests, the points it returns do not always generate a subgroup of full rank in the Mordell-Weil group, nor are they necessarily linearly independent. Moreover, the number of points can be smaller or larger than what one could expect after calling :meth:`~rank` or :meth:`~rank_bounds`.
.. NOTE::
The optional parameters control the Simon two descent algorithm; see the documentation of :meth:`~simon_two_descent` for more details.
INPUT:
- ``verbose`` -- 0, 1, 2, or 3 (default: 0), the verbosity level
- ``lim1`` -- (default: 2) limit on trivial points on quartics
- ``lim3`` -- (default: 4) limit on points on ELS quartics
- ``limtriv`` -- (default: 2) limit on trivial points on elliptic curve
- ``maxprob`` -- (default: 20)
- ``limbigprime`` -- (default: 30) to distinguish between small and large prime numbers. Use probabilistic tests for large primes. If 0, don't use probabilistic tests.
- ``known_points`` -- (default: None) list of known points on the curve
OUTPUT:
A set of points of infinite order given by the Simon two-descent.
.. NOTE::
For non-quadratic number fields, this code does return, but it takes a long time.
EXAMPLES::
sage: K.<a> = NumberField(x^2 + 23, 'a') sage: E = EllipticCurve(K,[0,0,0,101,0]) sage: E.gens() [(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)]
It can happen that no points are found if the height bounds used in the search are too small (see :trac:`10745`)::
sage: K.<y> = NumberField(x^4 + x^2 - 7) sage: E = EllipticCurve(K, [1, 0, 5*y^2 + 16, 0, 0]) sage: E.gens(lim1=1, lim3=1) [] sage: E.rank(), E.gens(lim3=12) # long time (about 4s) (1, [(369/25*y^3 + 539/25*y^2 + 1178/25*y + 1718/25 : -29038/125*y^3 - 43003/125*y^2 - 92706/125*y - 137286/125 : 1)])
Here is a curve of rank 2::
sage: K.<t> = NumberField(x^2-17) sage: E = EllipticCurve(K,[-4,0]) sage: E.gens() [(-1/2*t + 1/2 : -1/2*t + 1/2 : 1), (-t + 3 : -2*t + 10 : 1)] sage: E.rank() 2
Test that points of finite order are not included (see :trac:`13593`)::
sage: E = EllipticCurve("17a3") sage: K.<t> = NumberField(x^2+3) sage: EK = E.base_extend(K) sage: EK.rank() 0 sage: EK.gens() []
IMPLEMENTATION:
For curves over quadratic fields which are base-changes from `\QQ`, we delegate the work to :meth:`gens_quadratic` where methods over `\QQ` suffice. Otherwise, we use Denis Simon's PARI/GP scripts from http://www.math.unicaen.fr/~simon/. """
def period_lattice(self, embedding): r""" Returns the period lattice of the elliptic curve for the given embedding of its base field with respect to the differential `dx/(2y + a_1x + a_3)`.
INPUT:
- ``embedding`` - an embedding of the base number field into `\RR` or `\CC`.
.. note::
The precision of the embedding is ignored: we only use the given embedding to determine which embedding into ``QQbar`` to use. Once the lattice has been initialized, periods can be computed to arbitrary precision.
EXAMPLES:
First define a field with two real embeddings::
sage: K.<a> = NumberField(x^3-2) sage: E=EllipticCurve([0,0,0,a,2]) sage: embs=K.embeddings(CC); len(embs) 3
For each embedding we have a different period lattice::
sage: E.period_lattice(embs[0]) Period lattice associated to Elliptic Curve defined by y^2 = x^3 + a*x + 2 over Number Field in a with defining polynomial x^3 - 2 with respect to the embedding Ring morphism: From: Number Field in a with defining polynomial x^3 - 2 To: Algebraic Field Defn: a |--> -0.6299605249474365? - 1.091123635971722?*I
sage: E.period_lattice(embs[1]) Period lattice associated to Elliptic Curve defined by y^2 = x^3 + a*x + 2 over Number Field in a with defining polynomial x^3 - 2 with respect to the embedding Ring morphism: From: Number Field in a with defining polynomial x^3 - 2 To: Algebraic Field Defn: a |--> -0.6299605249474365? + 1.091123635971722?*I
sage: E.period_lattice(embs[2]) Period lattice associated to Elliptic Curve defined by y^2 = x^3 + a*x + 2 over Number Field in a with defining polynomial x^3 - 2 with respect to the embedding Ring morphism: From: Number Field in a with defining polynomial x^3 - 2 To: Algebraic Field Defn: a |--> 1.259921049894873?
Although the original embeddings have only the default precision, we can obtain the basis with higher precision later::
sage: L=E.period_lattice(embs[0]) sage: L.basis() (1.86405007647981 - 0.903761485143226*I, -0.149344633143919 - 2.06619546272945*I)
sage: L.basis(prec=100) (1.8640500764798108425920506200 - 0.90376148514322594749786960975*I, -0.14934463314391922099120107422 - 2.0661954627294548995621225062*I) """
def height_function(self): """ Return the canonical height function attached to self.
EXAMPLES::
sage: K.<a> = NumberField(x^2 - 5) sage: E = EllipticCurve(K, '11a3') sage: E.height_function() EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 over Number Field in a with defining polynomial x^2 - 5
"""
########################################################## # Isogeny class ########################################################## def isogeny_class(self): r""" Returns the isogeny class of this elliptic curve.
OUTPUT:
An instance of the class :class:`sage.schemes.elliptic_curves.isogeny_class.IsogenyClass_EC_NumberField`. From this object may be obtained a list of curves in the class, a matrix of the degrees of the isogenies between them, and the isogenies themselves.
.. note::
The curves in the isogeny class will all be minimal models if these exist (for example, when the class number is `1`); otherwise they will be minimal at all but one prime.
EXAMPLES::
sage: K.<i> = QuadraticField(-1) sage: E = EllipticCurve(K, [0,0,0,0,1]) sage: C = E.isogeny_class(); C Isogeny class of Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in i with defining polynomial x^2 + 1
The curves in the class (sorted)::
sage: [E1.ainvs() for E1 in C] [(0, 0, 0, 0, -27), (0, 0, 0, 0, 1), (i + 1, i, i + 1, -i + 3, 4*i), (i + 1, i, i + 1, -i + 33, -58*i)]
The matrix of degrees of cyclic isogenies between curves::
sage: C.matrix() [1 3 6 2] [3 1 2 6] [6 2 1 3] [2 6 3 1]
The array of isogenies themselves is not filled out but only contains those used to construct the class, the other entries containing the integer 0. This will be changed when the class :class:`EllipticCurveIsogeny` allowed composition. In this case we used `2`-isogenies to go from 0 to 2 and from 1 to 3, and `3`-isogenies to go from 0 to 1 and from 2 to 3::
sage: isogs = C.isogenies() sage: [((i,j),isogs[i][j].degree()) for i in range(4) for j in range(4) if isogs[i][j]!=0] [((0, 1), 3), ((0, 3), 2), ((1, 0), 3), ((1, 2), 2), ((2, 1), 2), ((2, 3), 3), ((3, 0), 2), ((3, 2), 3)] sage: [((i,j),isogs[i][j].x_rational_map()) for i in range(4) for j in range(4) if isogs[i][j]!=0] [((0, 1), (1/9*x^3 - 12)/x^2), ((0, 3), (-1/2*i*x^2 + i*x - 12*i)/(x - 3)), ((1, 0), (x^3 + 4)/x^2), ((1, 2), (-1/2*i*x^2 - i*x - 2*i)/(x + 1)), ((2, 1), (1/2*i*x^2 - x)/(x + 3/2*i)), ((2, 3), (x^3 + 4*i*x^2 - 10*x - 10*i)/(x^2 + 4*i*x - 4)), ((3, 0), (1/2*i*x^2 + x + 4*i)/(x - 5/2*i)), ((3, 2), (1/9*x^3 - 4/3*i*x^2 - 34/3*x + 226/9*i)/(x^2 - 8*i*x - 16))]
The isogeny class may be visualized by obtaining its graph and plotting it::
sage: G = C.graph() sage: G.show(edge_labels=True) # long time
sage: K.<i> = QuadraticField(-1) sage: E = EllipticCurve([1+i, -i, i, 1, 0]) sage: C = E.isogeny_class(); C Isogeny class of Elliptic Curve defined by y^2 + (i+1)*x*y + i*y = x^3 + (-i)*x^2 + x over Number Field in i with defining polynomial x^2 + 1 sage: len(C) 6 sage: C.matrix() [ 1 3 9 18 6 2] [ 3 1 3 6 2 6] [ 9 3 1 2 6 18] [18 6 2 1 3 9] [ 6 2 6 3 1 3] [ 2 6 18 9 3 1] sage: [E1.ainvs() for E1 in C] [(i + 1, i - 1, i, -i - 1, -i + 1), (i + 1, i - 1, i, 14*i + 4, 7*i + 14), (i + 1, i - 1, i, 59*i + 99, 372*i - 410), (i + 1, -i, i, -240*i - 399, 2869*i + 2627), (i + 1, -i, i, -5*i - 4, 2*i + 5), (i + 1, -i, i, 1, 0)]
An example with CM by `\sqrt{-5}`::
sage: pol = PolynomialRing(QQ,'x')([1,0,3,0,1]) sage: K.<c> = NumberField(pol) sage: j = 1480640+565760*c^2 sage: E = EllipticCurve(j=j) sage: E.has_cm() True sage: E.has_rational_cm() True sage: E.cm_discriminant() -20 sage: C = E.isogeny_class() sage: len(C) 2 sage: C.matrix() [1 2] [2 1] sage: [E.ainvs() for E in C] [(0, 0, 0, 83490*c^2 - 147015, -64739840*c^2 - 84465260), (0, 0, 0, -161535*c^2 + 70785, -62264180*c^3 + 6229080*c)] sage: C.isogenies()[0][1] Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (83490*c^2-147015)*x + (-64739840*c^2-84465260) over Number Field in c with defining polynomial x^4 + 3*x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + (-161535*c^2+70785)*x + (-62264180*c^3+6229080*c) over Number Field in c with defining polynomial x^4 + 3*x^2 + 1
An example with CM by `\sqrt{-23}` (class number `3`)::
sage: pol = PolynomialRing(QQ,'x')([1,-3,5,-5,5,-3,1]) sage: L.<a> = NumberField(pol) sage: js = hilbert_class_polynomial(-23).roots(L,multiplicities=False); len(js) 3 sage: E = EllipticCurve(j=js[0]) sage: E.has_rational_cm() True sage: len(E.isogenies_prime_degree()) 3 sage: C = E.isogeny_class(); len(C) 6
The reason for the isogeny class having size six while the class number is only `3` is that the class also contains three curves with CM by the order of discriminant `-92=4\cdot(-23)`, which also has class number `3`. The curves in the class are sorted first by CM discriminant (then lexicographically using a-invariants)::
sage: [F.cm_discriminant() for F in C] [-23, -23, -23, -92, -92, -92]
`2` splits in the order with discriminant `-23`, into two primes of order `3` in the class group, each of which induces a `2`-isogeny to a curve with the same endomorphism ring; the third `2`-isogeny is to a curve with the smaller endomorphism ring::
sage: [phi.codomain().cm_discriminant() for phi in E.isogenies_prime_degree()] [-92, -23, -23]
sage: C.matrix() [1 2 2 4 2 4] [2 1 2 2 4 4] [2 2 1 4 4 2] [4 2 4 1 3 3] [2 4 4 3 1 3] [4 4 2 3 3 1]
The graph of this isogeny class has a shape which does not occur over `\QQ`: a triangular prism. Note that for curves without CM, the graph has an edge between two curves if and only if they are connected by an isogeny of prime degree, and this degree is uniquely determined by the two curves, but in the CM case this property does not hold, since for pairs of curves in the class with the same endomorphism ring `O`, the set of degrees of isogenies between them is the set of integers represented by a primitive integral binary quadratic form of discriminant `\text{disc}(O)`, and this form represents infinitely many primes. In the matrix we give a small prime represented by the appropriate form. In this example, the matrix is formed by four `3\times3` blocks. The isogenies of degree `2` indicated by the upper left `3\times3` block of the matrix could be replaced by isogenies of any degree represented by the quadratic form `2x^2+xy+3y^2` of discriminant `-23`. Similarly in the lower right block, the entries of `3` could be represented by any integers represented by the quadratic form `3x^2+2xy+8y^2` of discriminant `-92`. In the top right block and lower left blocks, by contrast, the prime entries `2` are unique determined::
sage: G = C.graph() sage: G.adjacency_matrix() [0 1 1 0 1 0] [1 0 1 1 0 0] [1 1 0 0 0 1] [0 1 0 0 1 1] [1 0 0 1 0 1] [0 0 1 1 1 0]
To display the graph without any edge labels::
G.show() # long time
To display the graph with edge labels: by default, for curves with rational CM, the labels are the coefficients of the associated quadratic forms::
G.show(edge_labels=True) # long time
For an alternative view, first relabel the edges using only 2 labels to distinguish between isogenies between curves with the same endomorphism ring and isogenies between curves with different endomorphism rings, then use a 3-dimensional plot which can be rotated::
sage: for i,j,l in G.edge_iterator(): G.set_edge_label(i,j,l.count(',')) sage: G.show3d(color_by_label=True)
A class number `6` example. First we set up the fields: ``pol`` defines the same field as ``pol26`` but is simpler::
sage: pol26 = hilbert_class_polynomial(-4*26) sage: pol = x^6-x^5+2*x^4+x^3-2*x^2-x-1 sage: K.<a> = NumberField(pol) sage: L.<b> = K.extension(x^2+26)
Only `2` of the `j`-invariants with discriminant -104 are in `K`, though all are in `L`::
sage: len(pol26.roots(K)) 2 sage: len(pol26.roots(L)) 6
We create an elliptic curve defined over `K` with one of the `j`-invariants in `K`::
sage: j1 = pol26.roots(K)[0][0] sage: E = EllipticCurve(j=j1) sage: E.has_cm() True sage: E.has_rational_cm() False sage: E.has_rational_cm(L) True
Over `K` the isogeny class has size `4`, with `2` curves for each of the `2` `K`-rational `j`-invariants::
sage: C = E.isogeny_class(); len(C) # long time (~11s) 4 sage: C.matrix() # long time [ 1 13 2 26] [13 1 26 2] [ 2 26 1 13] [26 2 13 1] sage: len(Set([EE.j_invariant() for EE in C.curves])) # long time 2
Over `L`, the isogeny class grows to size `6` (the class number)::
sage: EL = E.change_ring(L) sage: CL = EL.isogeny_class(); len(CL) # long time (~80s) 6 sage: Set([EE.j_invariant() for EE in CL.curves]) == Set(pol26.roots(L,multiplicities=False)) # long time True
In each position in the matrix of degrees, we see primes (or `1`). In fact the set of degrees of cyclic isogenies from curve `i` to curve `j` is infinite, and is the set of all integers represented by one of the primitive binary quadratic forms of discriminant `-104`, from which we have selected a small prime::
sage: CL.matrix() # long time # random (see :trac:`19229`) [1 2 3 3 5 5] [2 1 5 5 3 3] [3 5 1 3 2 5] [3 5 3 1 5 2] [5 3 2 5 1 3] [5 3 5 2 3 1]
To see the array of binary quadratic forms::
sage: CL.qf_matrix() # long time # random (see :trac:`19229`) [[[1], [2, 0, 13], [3, -2, 9], [3, -2, 9], [5, -4, 6], [5, -4, 6]], [[2, 0, 13], [1], [5, -4, 6], [5, -4, 6], [3, -2, 9], [3, -2, 9]], [[3, -2, 9], [5, -4, 6], [1], [3, -2, 9], [2, 0, 13], [5, -4, 6]], [[3, -2, 9], [5, -4, 6], [3, -2, 9], [1], [5, -4, 6], [2, 0, 13]], [[5, -4, 6], [3, -2, 9], [2, 0, 13], [5, -4, 6], [1], [3, -2, 9]], [[5, -4, 6], [3, -2, 9], [5, -4, 6], [2, 0, 13], [3, -2, 9], [1]]]
As in the non-CM case, the isogeny class may be visualized by obtaining its graph and plotting it. Since there are more edges than in the non-CM case, it may be preferable to omit the edge_labels::
sage: G = C.graph() sage: G.show(edge_labels=False) # long time
It is possible to display a 3-dimensional plot, with colours to represent the different edge labels, in a form which can be rotated!::
sage: G.show3d(color_by_label=True) # long time
TESTS:
An example which failed until fixed at :trac:`19229`::
sage: K.<a> = NumberField(x^2-x+1) sage: E = EllipticCurve([a+1,1,1,0,0]) sage: C = E.isogeny_class(); len(C) 4 """
def isogenies_prime_degree(self, l=None): r""" Returns a list of `\ell`-isogenies from self, where `\ell` is a prime.
INPUT:
- ``l`` -- either None or a prime or a list of primes.
OUTPUT:
(list) `\ell`-isogenies for the given `\ell` or if `\ell` is None, all isogenies of prime degree (see below for the CM case).
.. note::
Over `\QQ`, the codomains of the isogenies returned are standard minimal models. Over other number fields they are global minimal models if these exist, otherwise models which are minimal at all but one prime.
.. note::
For curves with rational CM, isogenies of primes degree exist for infinitely many primes `\ell`, though there are only finitely many isogenous curves up to isomorphism. The list returned only includes one isogeny of prime degree for each codomain.
EXAMPLES::
sage: K.<i> = QuadraticField(-1) sage: E = EllipticCurve(K, [0,0,0,0,1]) sage: isogs = E.isogenies_prime_degree() sage: [phi.degree() for phi in isogs] [2, 3]
sage: pol = PolynomialRing(QQ,'x')([1,-3,5,-5,5,-3,1]) sage: L.<a> = NumberField(pol) sage: js = hilbert_class_polynomial(-23).roots(L,multiplicities=False); len(js) 3 sage: E = EllipticCurve(j=js[0]) sage: len(E.isogenies_prime_degree()) 3
TESTS::
sage: E.isogenies_prime_degree(4) Traceback (most recent call last): ... ValueError: 4 is not prime.
"""
except TypeError: raise ValueError("%s is not a prime integer" % l) else: raise ValueError("%s is not prime." % l)
# eliminate any endomorphisms and repeated codomains if not any([E.is_isomorphic(codoms[i]) for E in codoms[:i]])]
def is_isogenous(self, other, proof=True, maxnorm=100): """ Returns whether or not self is isogenous to other.
INPUT:
- ``other`` -- another elliptic curve.
- ``proof`` (default True) -- If ``False``, the function will return ``True`` whenever the two curves have the same conductor and are isogenous modulo `p` for all primes `p` of norm up to ``maxnorm``. If ``True``, the function returns False when the previous condition does not hold, and if it does hold we compute the complete isogeny class to see if the curves are indeed isogenous.
- ``maxnorm`` (integer, default 100) -- The maximum norm of primes `p` for which isogeny modulo `p` will be checked.
OUTPUT:
(bool) True if there is an isogeny from curve ``self`` to curve ``other``.
EXAMPLES::
sage: x = polygen(QQ, 'x') sage: F = NumberField(x^2 -2, 's'); F Number Field in s with defining polynomial x^2 - 2 sage: E1 = EllipticCurve(F, [7,8]) sage: E2 = EllipticCurve(F, [0,5,0,1,0]) sage: E3 = EllipticCurve(F, [0,-10,0,21,0]) sage: E1.is_isogenous(E2) False sage: E1.is_isogenous(E1) True sage: E2.is_isogenous(E2) True sage: E2.is_isogenous(E1) False sage: E2.is_isogenous(E3) True
::
sage: x = polygen(QQ, 'x') sage: F = NumberField(x^2 -2, 's'); F Number Field in s with defining polynomial x^2 - 2 sage: E = EllipticCurve('14a1') sage: EE = EllipticCurve('14a2') sage: E1 = E.change_ring(F) sage: E2 = EE.change_ring(F) sage: E1.is_isogenous(E2) True
::
sage: x = polygen(QQ, 'x') sage: F = NumberField(x^2 -2, 's'); F Number Field in s with defining polynomial x^2 - 2 sage: k.<a> = NumberField(x^3+7) sage: E = EllipticCurve(F, [7,8]) sage: EE = EllipticCurve(k, [2, 2]) sage: E.is_isogenous(EE) Traceback (most recent call last): ... ValueError: Second argument must be defined over the same number field.
Some examples from Cremona's 1981 tables::
sage: K.<i> = QuadraticField(-1) sage: E1 = EllipticCurve([i + 1, 0, 1, -240*i - 400, -2869*i - 2627]) sage: E1.conductor() Fractional ideal (-4*i - 7) sage: E2 = EllipticCurve([1+i,0,1,0,0]) sage: E2.conductor() Fractional ideal (-4*i - 7) sage: E1.is_isogenous(E2) # slower (~500ms) True sage: E1.is_isogenous(E2, proof=False) # faster (~170ms) True
In this case E1 and E2 are in fact 9-isogenous, as may be deduced from the following::
sage: E3 = EllipticCurve([i + 1, 0, 1, -5*i - 5, -2*i - 5]) sage: E3.is_isogenous(E1) True sage: E3.is_isogenous(E2) True sage: E1.isogeny_degree(E2) 9
TESTS:
Check that :trac:`15890` is fixed::
sage: K.<s> = QuadraticField(229) sage: c4 = 2173 - 235*(1 - s)/2 sage: c6 = -124369 + 15988*(1 - s)/2 sage: c4c = 2173 - 235*(1 + s)/2 sage: c6c = -124369 + 15988*(1 + s)/2 sage: E = EllipticCurve_from_c4c6(c4, c6) sage: Ec = EllipticCurve_from_c4c6(c4c, c6c) sage: E.is_isogenous(Ec) True
Check that :trac:`17295` is fixed::
sage: k.<s> = QuadraticField(2) sage: K.<b> = k.extension(x^2 - 3) sage: E = EllipticCurve(k, [-3*s*(4 + 5*s), 2*s*(2 + 14*s + 11*s^2)]) sage: Ec = EllipticCurve(k, [3*s*(4 - 5*s), -2*s*(2 - 14*s + 11*s^2)]) sage: EK = E.base_extend(K) sage: EcK = Ec.base_extend(K) sage: EK.is_isogenous(EcK) # long time (about 3.5 s) True
""" raise ValueError("Second argument is not an Elliptic Curve.")
return False
# We first try the easiest cases: primes for which X_0(l) has genus 0:
# Next we try the primes for which X_0^+(l) has genus 0 for # which isogeny-finding is faster than in general:
return True
# Next we try looking modulo some more primes:
return False
# Finally we compute the full isogeny class of E1 and check if # E2 is isomorphic to any curve in the class:
raise NotImplementedError("Curves appear to be isogenous (same conductor, isogenous modulo all primes of norm up to %s), but no isogeny has been constructed." % (10*maxnorm))
def isogeny_degree(self, other): """ Returns the minimal degree of an isogeny between self and other, or 0 if no isogeny exists.
INPUT:
- ``other`` -- another elliptic curve.
OUTPUT:
(int) The degree of an isogeny from ``self`` to ``other``, or 0.
EXAMPLES::
sage: x = QQ['x'].0 sage: F = NumberField(x^2 -2, 's'); F Number Field in s with defining polynomial x^2 - 2 sage: E = EllipticCurve('14a1') sage: EE = EllipticCurve('14a2') sage: E1 = E.change_ring(F) sage: E2 = EE.change_ring(F) sage: E1.isogeny_degree(E2) 2 sage: E2.isogeny_degree(E2) 1 sage: E5 = EllipticCurve('14a5').change_ring(F) sage: E1.isogeny_degree(E5) 6
sage: E = EllipticCurve('11a1') sage: [E2.label() for E2 in cremona_curves([11..20]) if E.isogeny_degree(E2)] ['11a1', '11a2', '11a3']
sage: K.<i> = QuadraticField(-1) sage: E = EllipticCurve([1+i, -i, i, 1, 0]) sage: C = E.isogeny_class() sage: [E.isogeny_degree(F) for F in C] [2, 6, 18, 9, 3, 1] """ # First deal with some easy cases: return Integer(0)
except ValueError: return ZZ(0)
def lll_reduce(self, points, height_matrix=None, precision=None): """ Returns an LLL-reduced basis from a given basis, with transform matrix.
INPUT:
- ``points`` - a list of points on this elliptic curve, which should be independent.
- ``height_matrix`` - the height-pairing matrix of the points, or ``None``. If ``None``, it will be computed.
- ``precision`` - number of bits of precision of intermediate computations (default: ``None``, for default RealField precision; ignored if ``height_matrix`` is supplied)
OUTPUT: A tuple (newpoints, U) where U is a unimodular integer matrix, new_points is the transform of points by U, such that new_points has LLL-reduced height pairing matrix
.. note::
If the input points are not independent, the output depends on the undocumented behaviour of PARI's ``qflllgram()`` function when applied to a gram matrix which is not positive definite.
EXAMPLES:
Some examples over `\QQ`::
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)] sage: Qi, U = E.lll_reduce(Pi) sage: all(sum(U[i,j]*Pi[i] for i in range(4)) == Qi[j] for j in range(4)) True sage: sorted(Qi) [(-4 : 1 : 1), (-3 : 5 : 1), (-2 : 6 : 1), (0 : 6 : 1)] sage: U.det() 1 sage: E.regulator_of_points(Pi) 4.59088036960573 sage: E.regulator_of_points(Qi) 4.59088036960574
::
sage: E = EllipticCurve([1,0,1,-120039822036992245303534619191166796374,504224992484910670010801799168082726759443756222911415116]) sage: xi = [2005024558054813068,\ -4690836759490453344,\ 4700156326649806635,\ 6785546256295273860,\ 6823803569166584943,\ 7788809602110240789,\ 27385442304350994620556,\ 54284682060285253719/4,\ -94200235260395075139/25,\ -3463661055331841724647/576,\ -6684065934033506970637/676,\ -956077386192640344198/2209,\ -27067471797013364392578/2809,\ -25538866857137199063309/3721,\ -1026325011760259051894331/108241,\ 9351361230729481250627334/1366561,\ 10100878635879432897339615/1423249,\ 11499655868211022625340735/17522596,\ 110352253665081002517811734/21353641,\ 414280096426033094143668538257/285204544,\ 36101712290699828042930087436/4098432361,\ 45442463408503524215460183165/5424617104,\ 983886013344700707678587482584/141566320009,\ 1124614335716851053281176544216033/152487126016] sage: points = [E.lift_x(x) for x in xi] sage: newpoints, U = E.lll_reduce(points) # long time (35s on sage.math, 2011) sage: [P[0] for P in newpoints] # long time [6823803569166584943, 5949539878899294213, 2005024558054813068, 5864879778877955778, 23955263915878682727/4, 5922188321411938518, 5286988283823825378, 175620639884534615751/25, -11451575907286171572, 3502708072571012181, 1500143935183238709184/225, 27180522378120223419/4, -5811874164190604461581/625, 26807786527159569093, 7404442636649562303, 475656155255883588, 265757454726766017891/49, 7272142121019825303, 50628679173833693415/4, 6951643522366348968, 6842515151518070703, 111593750389650846885/16, 2607467890531740394315/9, -1829928525835506297]
An example to show the explicit use of the height pairing matrix::
sage: E = EllipticCurve([0, 1, 1, -2, 42]) sage: Pi = E.gens() sage: H = E.height_pairing_matrix(Pi,3) sage: E.lll_reduce(Pi,height_matrix=H) ( [1 0 0 1] [0 1 0 1] [0 0 0 1] [(-4 : 1 : 1), (-3 : 5 : 1), (-2 : 6 : 1), (1 : -7 : 1)], [0 0 1 1] )
Some examples over number fields (see :trac:`9411`)::
sage: K.<a> = QuadraticField(-23, 'a') sage: E = EllipticCurve(K, [0,0,1,-1,0]) sage: P = E(-2,-(a+1)/2) sage: Q = E(0,-1) sage: E.lll_reduce([P,Q]) ( [0 1] [(0 : -1 : 1), (-2 : -1/2*a - 1/2 : 1)], [1 0] )
::
sage: K.<a> = QuadraticField(-5) sage: E = EllipticCurve(K,[0,a]) sage: points = [E.point([-211/841*a - 6044/841,-209584/24389*a + 53634/24389]),E.point([-17/18*a - 1/9, -109/108*a - 277/108]) ] sage: E.lll_reduce(points) ( [(-a + 4 : -3*a + 7 : 1), (-17/18*a - 1/9 : 109/108*a + 277/108 : 1)], [ 1 0] [ 1 -1] ) """ for i in range(r)]
def galois_representation(self): r""" The compatible family of the Galois representation attached to this elliptic curve.
Given an elliptic curve `E` over a number field `K` and a rational prime number `p`, the `p^n`-torsion `E[p^n]` points of `E` is a representation of the absolute Galois group of `K`. As `n` varies we obtain the Tate module `T_p E` which is a a representation of `G_K` on a free `\ZZ_p`-module of rank `2`. As `p` varies the representations are compatible.
EXAMPLES::
sage: K = NumberField(x**2 + 1, 'a') sage: E = EllipticCurve('11a1').change_ring(K) sage: rho = E.galois_representation() sage: rho Compatible family of Galois representations associated to the Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in a with defining polynomial x^2 + 1 sage: rho.is_surjective(3) True sage: rho.is_surjective(5) # long time (4s on sage.math, 2014) False sage: rho.non_surjective() [5] """
@cached_method def cm_discriminant(self): """ Returns the CM discriminant of the `j`-invariant of this curve, or 0.
OUTPUT:
An integer `D` which is either `0` if this curve `E` does not have Complex Multiplication) (CM), or an imaginary quadratic discriminant if `j(E)` is the `j`-invariant of the order with discriminant `D`.
.. note::
If `E` has CM but the discriminant `D` is not a square in the base field `K` then the extra endomorphisms will not be defined over `K`. See also :meth:`has_rational_cm`.
EXAMPLES::
sage: EllipticCurve(j=0).cm_discriminant() -3 sage: EllipticCurve(j=1).cm_discriminant() Traceback (most recent call last): ... ValueError: Elliptic Curve defined by y^2 + x*y = x^3 + 36*x + 3455 over Rational Field does not have CM sage: EllipticCurve(j=1728).cm_discriminant() -4 sage: EllipticCurve(j=8000).cm_discriminant() -8 sage: K.<a> = QuadraticField(5) sage: EllipticCurve(j=282880*a + 632000).cm_discriminant() -20 sage: K.<a> = NumberField(x^3 - 2) sage: EllipticCurve(j=31710790944000*a^2 + 39953093016000*a + 50337742902000).cm_discriminant() -108 """ else: # no CM
@cached_method def has_cm(self): """ Returns whether or not this curve has a CM `j`-invariant.
OUTPUT:
``True`` if this curve has CM over the algebraic closure of the base field, otherwise ``False``. See also :meth:`cm_discriminant()` and :meth:`has_rational_cm`.
.. note::
Even if `E` has CM in this sense (that its `j`-invariant is a CM `j`-invariant), if the associated negative discriminant `D` is not a square in the base field `K`, the extra endomorphisms will not be defined over `K`. See also the method :meth:`has_rational_cm` which tests whether `E` has extra endomorphisms defined over `K` or a given extension of `K`.
EXAMPLES::
sage: EllipticCurve(j=0).has_cm() True sage: EllipticCurve(j=1).has_cm() False sage: EllipticCurve(j=1728).has_cm() True sage: EllipticCurve(j=8000).has_cm() True sage: K.<a> = QuadraticField(5) sage: EllipticCurve(j=282880*a + 632000).has_cm() True sage: K.<a> = NumberField(x^3 - 2) sage: EllipticCurve(j=31710790944000*a^2 + 39953093016000*a + 50337742902000).has_cm() True """
@cached_method def has_rational_cm(self, field=None): """ Returns whether or not this curve has CM defined over its base field or a given extension.
INPUT:
- ``field`` -- a field, which should be an extension of the base field of the curve. If ``field`` is ``None`` (the default), it is taken to be the base field of the curve.
OUTPUT:
``True`` if the ring of endomorphisms of this curve over the given field is larger than `\ZZ`; otherwise ``False``. See also :meth:`cm_discriminant()` and :meth:`has_cm`.
.. note::
If `E` has CM but the discriminant `D` is not a square in the given field `K` then the extra endomorphisms will not be defined over `K`, and this function will return ``False``. See also :meth:`has_cm`. To obtain the CM discriminant, use :meth:`cm_discriminant()`.
EXAMPLES::
sage: E = EllipticCurve(j=0) sage: E.has_cm() True sage: E.has_rational_cm() False sage: D = E.cm_discriminant(); D -3 sage: E.has_rational_cm(QuadraticField(D)) True
sage: E = EllipticCurve(j=1728) sage: E.has_cm() True sage: E.has_rational_cm() False sage: D = E.cm_discriminant(); D -4 sage: E.has_rational_cm(QuadraticField(D)) True
Higher degree examples::
sage: K.<a> = QuadraticField(5) sage: E = EllipticCurve(j=282880*a + 632000) sage: E.has_cm() True sage: E.has_rational_cm() False sage: E.cm_discriminant() -20 sage: E.has_rational_cm(K.extension(x^2+5,'b')) True
An error is raised if a field is given which is not an extension of the base field::
sage: E.has_rational_cm(QuadraticField(-20)) Traceback (most recent call last): ... ValueError: Error in has_rational_cm: Number Field in a with defining polynomial x^2 + 20 is not an extension field of Number Field in a with defining polynomial x^2 - 5
sage: K.<a> = NumberField(x^3 - 2) sage: E = EllipticCurve(j=31710790944000*a^2 + 39953093016000*a + 50337742902000) sage: E.has_cm() True sage: E.has_rational_cm() False sage: D = E.cm_discriminant(); D -108 sage: E.has_rational_cm(K.extension(x^2+108,'b')) True """ % (field,self.base_field()))
def saturation(self, points, verbose=False, max_prime=0, one_prime=0, odd_primes_only=False, lower_ht_bound=None, reg=None, debug=False): r""" Given a list of rational points on `E` over `K`, compute the saturation in `E(K)` of the subgroup they generate.
INPUT:
- ``points (list)`` - list of points on E. Points of finite order are ignored; the remaining points should be independent, or an error is raised.
- ``verbose`` (bool) - (default: ``False``), if ``True``, give verbose output.
- ``max_prime`` (int, default 0), saturation is performed for all primes up to ``max_prime``. If ``max_prime`` is 0, perform saturation at *all* primes, i.e., compute the true saturation.
- ``odd_primes_only`` (bool, default ``False``) -- only do saturation at odd primes.
- ``one_prime`` (int, default 0) -- if nonzero, only do saturation at this prime.
The following two inputs are optional, and may be provided to speed up the computation.
- ``lower_ht_bound`` (real, default ``None``) -- lower bound of the regulator `E(K)`, if known.
- ``reg`` (real, default ``None``), regulator of the span of points, if known.
- ``debug`` (int, default 0) -- , used for debugging and testing.
OUTPUT:
- ``saturation`` (list) - points that form a basis for the saturation.
- ``index`` (int) - the index of the group generated by the input points in their saturation.
- ``regulator`` (real with default precision, or ``None``) - regulator of saturated points.
EXAMPLES::
sage: K.<i> = QuadraticField(-1) sage: E = EllipticCurve('389a1') sage: EK = E.change_ring(K) sage: P = EK(-1,1); Q = EK(0,-1)
sage: EK.saturation([2*P], max_prime=2) ([(-1 : 1 : 1)], 2, 0.686667083305587) sage: EK.saturation([12*P], max_prime=2) ([(26/361 : -5720/6859 : 1)], 4, 6.18000374975028) sage: EK.saturation([12*P], lower_ht_bound=0.1) ([(-1 : 1 : 1)], 12, 0.686667083305587) sage: EK.saturation([2*P, Q], max_prime=2) ([(-1 : 1 : 1), (0 : -1 : 1)], 2, 0.152460177943144) sage: EK.saturation([P+Q, P-Q], lower_ht_bound=.1, debug=2) ([(-1 : 1 : 1), (1 : 0 : 1)], 2, 0.152460177943144) sage: EK.saturation([P+Q, 17*Q], lower_ht_bound=0.1) ([(4 : 8 : 1), (0 : -1 : 1)], 17, 0.152460177943143)
sage: R = EK(i-2,-i-3) sage: EK.saturation([P+R, P+Q, Q+R], lower_ht_bound=0.1) ([(841/1369*i - 171/1369 : 41334/50653*i - 74525/50653 : 1), (4 : 8 : 1), (-1/25*i + 18/25 : -69/125*i - 58/125 : 1)], 2, 0.103174443217351) sage: EK.saturation([26*Q], lower_ht_bound=0.1) ([(0 : -1 : 1)], 26, 0.327000773651605)
Another number field::
sage: E = EllipticCurve('389a1') sage: K.<a> = NumberField(x^3-x+1) sage: EK = E.change_ring(K) sage: P = EK(-1,1); Q = EK(0,-1) sage: EK.saturation([P+Q, P-Q], lower_ht_bound=0.1) ([(-1 : 1 : 1), (1 : 0 : 1)], 2, 0.152460177943144) sage: EK.saturation([3*P, P+5*Q], lower_ht_bound=0.1) ([(-185/2209 : -119510/103823 : 1), (80041/34225 : -26714961/6331625 : 1)], 15, 0.152460177943144)
A different curve::
sage: K.<a> = QuadraticField(3) sage: E = EllipticCurve('37a1') sage: EK = E.change_ring(K) sage: P = EK(0,0); Q = EK(2-a,2*a-4) sage: EK.saturation([3*P-Q, 3*P+Q], lower_ht_bound=.01) ([(0 : 0 : 1), (1/2*a : -1/4*a - 1/4 : 1)], 6, 0.0317814530725985)
The points must be linearly independent::
sage: EK.saturation([2*P, 3*Q, P-Q]) Traceback (most recent call last): ... ValueError: points not linearly independent in saturation()
Degenerate case::
sage: EK.saturation([]) ([], 1, 1.00000000000000)
ALGORITHM:
For rank 1 subgroups, simply do trial division up to the maximal prime divisor. For higher rank subgroups, perform trial division on all linear combinations for small primes, and look for projections `E(K) \rightarrow \oplus E(k) \otimes \mathbf{F}_p` which are either full rank or provide `p`-divisible linear combinations, where the `k` here are residue fields of `K`.
TESTS::
sage: K.<i> = QuadraticField(-1) sage: E = EllipticCurve('389a1') sage: EK = E.change_ring(K) sage: P = EK(-1,1); Q = EK(0,-1)
sage: EK.saturation([P+Q, P-Q], lower_ht_bound=.1, debug=2) ([(-1 : 1 : 1), (1 : 0 : 1)], 2, 0.152460177943144) sage: EK.saturation([5*P+6*Q, 5*P-3*Q], lower_ht_bound=.1) ([(-3/4 : -15/8 : 1), (159965/16129 : -67536260/2048383 : 1)], 45, 0.152460177943144) sage: EK.saturation([5*P+6*Q, 5*P-3*Q], lower_ht_bound=.1, debug=2) ([(-3/4 : -15/8 : 1), (159965/16129 : -67536260/2048383 : 1)], 45, 0.152460177943144) """
# compute the list of primes p at which p-saturation is # required.
# TODO (robertb): verify this for rank > 1 if verbose: print("Computing lower height bound..") lower_ht_bound = self.height_function().min(.1, 5) ** n if verbose: print("..done: %s" % lower_ht_bound) print("Testing primes up to %s" % prime_list[-1]) else: prime_list = [one_prime] else: prime_list.remove(2)
# Now saturate at each prime in prime_list. The dict # lin_combs keeps the values of linear combinations of the # points, indexed by coefficient tuples, for efficiency; it is # rest whenever the point list changes.
print("Saturating at p=%s" % p) print(" --gaining index %s^%s" % (p,expo)) else: print(" --already %s-saturated" % p)
def gens_quadratic(self, **kwds): """ Return generators for the Mordell-Weil group modulo torsion, for a curve which is a base change from `\QQ` to a quadratic field.
EXAMPLES::
sage: E = EllipticCurve([1,2,3,40,50]) sage: E.conductor() 2123582 sage: E.gens() [(5 : 17 : 1)] sage: K.<i> = QuadraticField(-1) sage: EK = E.change_ring(K) sage: EK.gens_quadratic() [(5 : 17 : 1), (-13 : 48*i + 5 : 1)]
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)]
sage: K.<a> = QuadraticField(-7) sage: E = EllipticCurve([0,0,0,197,0]) sage: E.conductor() 2483776 sage: E.gens() [(47995604297578081/7389879786648100 : -25038161802544048018837479/635266655830129794121000 : 1)] sage: K.<a> = QuadraticField(7) sage: E.change_ring(K).gens_quadratic() [(-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() Traceback (most recent call last): ... ValueError: gens_quadratic() requires the elliptic curve to be a base change from Q
"""
# In all cases there are exactly two distinct curves /Q whose # base-change to K is the original. NB These need not be # quadratic twists of each other! For example, '32a1' and # '32a2' are not quadratic twists of each other (each is its # own twist by -1) but they become isomorphic over # Q(sqrt(-1)).
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