Torsion subgroups of elliptic curves over number fields (including \(\QQ\))¶
AUTHORS:
Nick Alexander: original implementation over \(\QQ\)
Chris Wuthrich: original implementation over number fields
John Cremona: rewrote p-primary part to use division polynomials, added some features, unified Number Field and \(\QQ\) code.
- class sage.schemes.elliptic_curves.ell_torsion.EllipticCurveTorsionSubgroup(E)[source]¶
Bases:
AdditiveAbelianGroupWrapperThe torsion subgroup of an elliptic curve over a number field.
EXAMPLES:
Examples over \(\QQ\):
sage: E = EllipticCurve([-4, 0]); E Elliptic Curve defined by y^2 = x^3 - 4*x over Rational Field sage: G = E.torsion_subgroup(); G Torsion Subgroup isomorphic to Z/2 + Z/2 associated to the Elliptic Curve defined by y^2 = x^3 - 4*x over Rational Field sage: G.order() 4 sage: G.gen(0) (-2 : 0 : 1) sage: G.gen(1) (0 : 0 : 1) sage: G.ngens() 2
>>> from sage.all import * >>> E = EllipticCurve([-Integer(4), Integer(0)]); E Elliptic Curve defined by y^2 = x^3 - 4*x over Rational Field >>> G = E.torsion_subgroup(); G Torsion Subgroup isomorphic to Z/2 + Z/2 associated to the Elliptic Curve defined by y^2 = x^3 - 4*x over Rational Field >>> G.order() 4 >>> G.gen(Integer(0)) (-2 : 0 : 1) >>> G.gen(Integer(1)) (0 : 0 : 1) >>> G.ngens() 2
E = EllipticCurve([-4, 0]); E G = E.torsion_subgroup(); G G.order() G.gen(0) G.gen(1) G.ngens()
sage: E = EllipticCurve([17, -120, -60, 0, 0]); E Elliptic Curve defined by y^2 + 17*x*y - 60*y = x^3 - 120*x^2 over Rational Field sage: G = E.torsion_subgroup(); G Torsion Subgroup isomorphic to Trivial group associated to the Elliptic Curve defined by y^2 + 17*x*y - 60*y = x^3 - 120*x^2 over Rational Field sage: G.gens() () sage: e = EllipticCurve([0, 33076156654533652066609946884, 0, ....: 347897536144342179642120321790729023127716119338758604800, ....: 1141128154369274295519023032806804247788154621049857648870032370285851781352816640000]) sage: e.torsion_order() 16
>>> from sage.all import * >>> E = EllipticCurve([Integer(17), -Integer(120), -Integer(60), Integer(0), Integer(0)]); E Elliptic Curve defined by y^2 + 17*x*y - 60*y = x^3 - 120*x^2 over Rational Field >>> G = E.torsion_subgroup(); G Torsion Subgroup isomorphic to Trivial group associated to the Elliptic Curve defined by y^2 + 17*x*y - 60*y = x^3 - 120*x^2 over Rational Field >>> G.gens() () >>> e = EllipticCurve([Integer(0), Integer(33076156654533652066609946884), Integer(0), ... Integer(347897536144342179642120321790729023127716119338758604800), ... Integer(1141128154369274295519023032806804247788154621049857648870032370285851781352816640000)]) >>> e.torsion_order() 16
E = EllipticCurve([17, -120, -60, 0, 0]); E G = E.torsion_subgroup(); G G.gens() e = EllipticCurve([0, 33076156654533652066609946884, 0, 347897536144342179642120321790729023127716119338758604800, 1141128154369274295519023032806804247788154621049857648870032370285851781352816640000]) e.torsion_order()>>> from sage.all import * >>> E = EllipticCurve([Integer(17), -Integer(120), -Integer(60), Integer(0), Integer(0)]); E Elliptic Curve defined by y^2 + 17*x*y - 60*y = x^3 - 120*x^2 over Rational Field >>> G = E.torsion_subgroup(); G Torsion Subgroup isomorphic to Trivial group associated to the Elliptic Curve defined by y^2 + 17*x*y - 60*y = x^3 - 120*x^2 over Rational Field >>> G.gens() () >>> e = EllipticCurve([Integer(0), Integer(33076156654533652066609946884), Integer(0), ... Integer(347897536144342179642120321790729023127716119338758604800), ... Integer(1141128154369274295519023032806804247788154621049857648870032370285851781352816640000)]) >>> e.torsion_order() 16
E = EllipticCurve([17, -120, -60, 0, 0]); E G = E.torsion_subgroup(); G G.gens() e = EllipticCurve([0, 33076156654533652066609946884, 0, 347897536144342179642120321790729023127716119338758604800, 1141128154369274295519023032806804247788154621049857648870032370285851781352816640000]) e.torsion_order()Constructing points from the torsion subgroup:
sage: E = EllipticCurve('14a1') sage: T = E.torsion_subgroup() sage: [E(t) for t in T] [(0 : 1 : 0), (9 : 23 : 1), (2 : 2 : 1), (1 : -1 : 1), (2 : -5 : 1), (9 : -33 : 1)]
>>> from sage.all import * >>> E = EllipticCurve('14a1') >>> T = E.torsion_subgroup() >>> [E(t) for t in T] [(0 : 1 : 0), (9 : 23 : 1), (2 : 2 : 1), (1 : -1 : 1), (2 : -5 : 1), (9 : -33 : 1)]
E = EllipticCurve('14a1') T = E.torsion_subgroup() [E(t) for t in T]An example where the torsion subgroup is not cyclic:
sage: E = EllipticCurve([0,0,0,-49,0]) sage: T = E.torsion_subgroup() sage: [E(t) for t in T] [(0 : 1 : 0), (0 : 0 : 1), (-7 : 0 : 1), (7 : 0 : 1)]
>>> from sage.all import * >>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),-Integer(49),Integer(0)]) >>> T = E.torsion_subgroup() >>> [E(t) for t in T] [(0 : 1 : 0), (0 : 0 : 1), (-7 : 0 : 1), (7 : 0 : 1)]
E = EllipticCurve([0,0,0,-49,0]) T = E.torsion_subgroup() [E(t) for t in T]
An example where the torsion subgroup is trivial:
sage: E = EllipticCurve('37a1') sage: T = E.torsion_subgroup() sage: T Torsion Subgroup isomorphic to Trivial group associated to the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field sage: [E(t) for t in T] [(0 : 1 : 0)]
>>> from sage.all import * >>> E = EllipticCurve('37a1') >>> T = E.torsion_subgroup() >>> T Torsion Subgroup isomorphic to Trivial group associated to the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field >>> [E(t) for t in T] [(0 : 1 : 0)]
E = EllipticCurve('37a1') T = E.torsion_subgroup() T [E(t) for t in T]Examples over other Number Fields:
sage: # needs sage.rings.number_field sage: E = EllipticCurve('11a1') sage: x = polygen(ZZ, 'x') sage: K.<i> = NumberField(x^2 + 1) sage: EK = E.change_ring(K) sage: from sage.schemes.elliptic_curves.ell_torsion import EllipticCurveTorsionSubgroup sage: EllipticCurveTorsionSubgroup(EK) Torsion Subgroup isomorphic to Z/5 associated to the Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in i with defining polynomial x^2 + 1 sage: E = EllipticCurve('11a1') sage: K.<i> = NumberField(x^2 + 1) # needs sage.rings.number_field sage: EK = E.change_ring(K) # needs sage.rings.number_field sage: T = EK.torsion_subgroup() # needs sage.rings.number_field sage: T.ngens() 1 sage: T.gen(0) (5 : -6 : 1)
>>> from sage.all import * >>> # needs sage.rings.number_field >>> E = EllipticCurve('11a1') >>> x = polygen(ZZ, 'x') >>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1) >>> EK = E.change_ring(K) >>> from sage.schemes.elliptic_curves.ell_torsion import EllipticCurveTorsionSubgroup >>> EllipticCurveTorsionSubgroup(EK) Torsion Subgroup isomorphic to Z/5 associated to the Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in i with defining polynomial x^2 + 1 >>> E = EllipticCurve('11a1') >>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)# needs sage.rings.number_field >>> EK = E.change_ring(K) # needs sage.rings.number_field >>> T = EK.torsion_subgroup() # needs sage.rings.number_field >>> T.ngens() 1 >>> T.gen(Integer(0)) (5 : -6 : 1)
# needs sage.rings.number_field E = EllipticCurve('11a1') x = polygen(ZZ, 'x') K.<i> = NumberField(x^2 + 1) EK = E.change_ring(K) from sage.schemes.elliptic_curves.ell_torsion import EllipticCurveTorsionSubgroup EllipticCurveTorsionSubgroup(EK) E = EllipticCurve('11a1') K.<i> = NumberField(x^2 + 1) # needs sage.rings.number_field EK = E.change_ring(K) # needs sage.rings.number_field T = EK.torsion_subgroup() # needs sage.rings.number_field T.ngens() T.gen(0)Note: this class is normally constructed indirectly as follows:
sage: # needs sage.rings.number_field sage: T = EK.torsion_subgroup(); T Torsion Subgroup isomorphic to Z/5 associated to the Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in i with defining polynomial x^2 + 1 sage: type(T) <class 'sage.schemes.elliptic_curves.ell_torsion.EllipticCurveTorsionSubgroup_with_category'>
>>> from sage.all import * >>> # needs sage.rings.number_field >>> T = EK.torsion_subgroup(); T Torsion Subgroup isomorphic to Z/5 associated to the Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in i with defining polynomial x^2 + 1 >>> type(T) <class 'sage.schemes.elliptic_curves.ell_torsion.EllipticCurveTorsionSubgroup_with_category'>
# needs sage.rings.number_field T = EK.torsion_subgroup(); T type(T)
AUTHORS:
Nick Alexander: initial implementation over \(\QQ\).
Chris Wuthrich: initial implementation over number fields.
John Cremona: additional features and unification.
- curve()[source]¶
Return the curve of this torsion subgroup.
EXAMPLES:
sage: # needs sage.rings.number_field sage: E = EllipticCurve('11a1') sage: x = polygen(ZZ, 'x') sage: K.<i> = NumberField(x^2 + 1) sage: EK = E.change_ring(K) sage: T = EK.torsion_subgroup() sage: T.curve() is EK True
>>> from sage.all import * >>> # needs sage.rings.number_field >>> E = EllipticCurve('11a1') >>> x = polygen(ZZ, 'x') >>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1) >>> EK = E.change_ring(K) >>> T = EK.torsion_subgroup() >>> T.curve() is EK True
# needs sage.rings.number_field E = EllipticCurve('11a1') x = polygen(ZZ, 'x') K.<i> = NumberField(x^2 + 1) EK = E.change_ring(K) T = EK.torsion_subgroup() T.curve() is EK
- points()[source]¶
Return a list of all the points in this torsion subgroup.
The list is cached.
EXAMPLES:
sage: # needs sage.rings.number_field sage: x = polygen(ZZ, 'x') sage: K.<i> = NumberField(x^2 + 1) sage: E = EllipticCurve(K, [0,0,0,1,0]) sage: tor = E.torsion_subgroup() sage: tor.points() [(0 : 1 : 0), (0 : 0 : 1), (-i : 0 : 1), (i : 0 : 1)]
>>> from sage.all import * >>> # needs sage.rings.number_field >>> x = polygen(ZZ, 'x') >>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1) >>> E = EllipticCurve(K, [Integer(0),Integer(0),Integer(0),Integer(1),Integer(0)]) >>> tor = E.torsion_subgroup() >>> tor.points() [(0 : 1 : 0), (0 : 0 : 1), (-i : 0 : 1), (i : 0 : 1)]
# needs sage.rings.number_field x = polygen(ZZ, 'x') K.<i> = NumberField(x^2 + 1) E = EllipticCurve(K, [0,0,0,1,0]) tor = E.torsion_subgroup() tor.points()
- sage.schemes.elliptic_curves.ell_torsion.torsion_bound(E, number_of_places=20)[source]¶
Return an upper bound on the order of the torsion subgroup.
INPUT:
E– an elliptic curve over \(\QQ\) or a number fieldnumber_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.
To avoid nontrivial arithmetic in the base field (in particular, to avoid having to compute the maximal order) we only use prime \(P\) above rational primes \(p\) which do not divide the discriminant of the equation order.
EXAMPLES:
sage: CDB = CremonaDatabase() sage: from sage.schemes.elliptic_curves.ell_torsion import torsion_bound sage: [torsion_bound(E) 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]
>>> from sage.all import * >>> CDB = CremonaDatabase() >>> from sage.schemes.elliptic_curves.ell_torsion import torsion_bound >>> [torsion_bound(E) for E in CDB.iter([Integer(14)])] [6, 6, 6, 6, 6, 6] >>> [E.torsion_order() for E in CDB.iter([Integer(14)])] [6, 6, 2, 6, 2, 6]
CDB = CremonaDatabase() from sage.schemes.elliptic_curves.ell_torsion import torsion_bound [torsion_bound(E) for E in CDB.iter([14])] [E.torsion_order() for E in CDB.iter([14])]
An example over a relative number field (see Issue #16011):
sage: # needs sage.rings.number_field 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
>>> from sage.all import * >>> # needs sage.rings.number_field >>> R = QQ['x']; (x,) = R._first_ngens(1) >>> F = QuadraticField(Integer(5), names=('a',)); (a,) = F._first_ngens(1) >>> K = F.extension(x**Integer(2) - Integer(3), names=('b',)); (b,) = K._first_ngens(1) >>> E = EllipticCurve(K, [Integer(0),Integer(0),Integer(0),b,Integer(1)]) >>> E.torsion_subgroup().order() 1
# needs sage.rings.number_field R.<x> = QQ[] F.<a> = QuadraticField(5) K.<b> = F.extension(x^2 - 3) E = EllipticCurve(K, [0,0,0,b,1]) E.torsion_subgroup().order()
An example of a base-change curve from \(\QQ\) to a degree 16 field:
sage: # needs sage.rings.number_field sage: from sage.schemes.elliptic_curves.ell_torsion import torsion_bound sage: f = PolynomialRing(QQ,'x')([5643417737593488384,0, ....: -11114515801179776,0,-455989850911004,0,379781901872, ....: 0,14339154953,0,-1564048,0,-194542,0,-32,0,1]) sage: K = NumberField(f,'a') sage: E = EllipticCurve(K, [1, -1, 1, 824579, 245512517]) sage: torsion_bound(E) 16 sage: E.torsion_subgroup().invariants() (4, 4)
>>> from sage.all import * >>> # needs sage.rings.number_field >>> from sage.schemes.elliptic_curves.ell_torsion import torsion_bound >>> f = PolynomialRing(QQ,'x')([Integer(5643417737593488384),Integer(0), ... -Integer(11114515801179776),Integer(0),-Integer(455989850911004),Integer(0),Integer(379781901872), ... Integer(0),Integer(14339154953),Integer(0),-Integer(1564048),Integer(0),-Integer(194542),Integer(0),-Integer(32),Integer(0),Integer(1)]) >>> K = NumberField(f,'a') >>> E = EllipticCurve(K, [Integer(1), -Integer(1), Integer(1), Integer(824579), Integer(245512517)]) >>> torsion_bound(E) 16 >>> E.torsion_subgroup().invariants() (4, 4)
# needs sage.rings.number_field from sage.schemes.elliptic_curves.ell_torsion import torsion_bound f = PolynomialRing(QQ,'x')([5643417737593488384,0, -11114515801179776,0,-455989850911004,0,379781901872, 0,14339154953,0,-1564048,0,-194542,0,-32,0,1]) K = NumberField(f,'a') E = EllipticCurve(K, [1, -1, 1, 824579, 245512517]) torsion_bound(E) E.torsion_subgroup().invariants()