Torsion subgroups of elliptic curves over number fields (including ).

AUTHORS:

• Nick Alexander: original implementation over

• Chris Wuthrich: original implementation over number fields

• John Cremona: rewrote p-primary part to use division

  polynomials, added some features, unified Number Field and code.

class sage.schemes.elliptic_curves.ell_torsion.EllipticCurveTorsionSubgroup(E, algorithm=None)

Bases: sage.groups.abelian_gps.abelian_group.AbelianGroup_class

The torsion subgroup of an elliptic curve over a number field.

EXAMPLES:

Examples over :

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 Multiplicative Abelian Group isomorphic to C2 x C2 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

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 Abelian 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

Constructing points from the torsion subgroup (which is an abstract abelian group):

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)]

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)]

An example where the torsion subgroup is trivial:

sage: E = EllipticCurve('37a1')
sage: T = E.torsion_subgroup()
sage: T
Torsion Subgroup isomorphic to Trivial Abelian 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)]

Examples over other Number Fields:

sage: E=EllipticCurve('11a1')
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 Multiplicative Abelian Group isomorphic to C5 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

Note: this class is normally constructed indirectly as follows:

sage: T = EK.torsion_subgroup(); T
Torsion Subgroup isomorphic to Multiplicative Abelian Group isomorphic to C5 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'>

AUTHORS:

• Nick Alexander - initial implementation over .
• Chris Wuthrich - initial implementation over number fields.
• John Cremona - additional features and unification.

curve()

Return the curve of this torsion subgroup.

EXAMPLES:

sage: E=EllipticCurve('11a1')
sage: K.<i>=NumberField(x^2+1)
sage: EK=E.change_ring(K)
sage: T = EK.torsion_subgroup()
sage: T.curve() is EK
True

gen(i=0)

Return the ‘th torsion generator.

EXAMPLES:

sage: E=EllipticCurve('11a1')
sage: K.<i>=NumberField(x^2+1)
sage: EK=E.change_ring(K)
sage: T = EK.torsion_subgroup()
sage: T.gen()
(16 : 60 : 1)

ngens()

Return the number of torsion generators.

EXAMPLES:

sage: E=EllipticCurve('11a1')
sage: K.<i>=NumberField(x^2+1)
sage: EK=E.change_ring(K)
sage: T = EK.torsion_subgroup()
sage: T.ngens()
1

points()

Return a list of all the points in this torsion subgroup. The list is cached.

EXAMPLES:

sage: K.<i>=NumberField(x^2 + 1)
sage: E = EllipticCurve(K,[0,0,0,1,0])
sage: tor = E.torsion_subgroup()
sage: tor.points()
[(i : 0 : 1), (0 : 0 : 1), (-i : 0 : 1), (0 : 1 : 0)]