The following are some general functions for working with

Subsections

• Creation of Subgroups of Elliptic Curves

A subgroup of an elliptic curve Es is a finite subscheme of E defined by a polynomial psi . The rational points of Es are those rational points of E whose x-coordinate is a root of psi . No checking is done to ensure that the rational points of Es do form a group under the addition law on E.

Finite subgroups of this type occur naturally in many places in the theory of elliptic curves. For instance, the kernel of an isogeny of elliptic curves has a natural subgroup structure. Any (finite) subgroup defines a unique separable isogeny in this way.

Subgroup(E, r) : CurveEll, RngUPolElt -> CurveEll

Creates the subgroup of the elliptic curve E defined by the univariate polynomial r.

Returns the order of the group of rational points of the subgroup. Subgroups in Magma are always finite.

Returns the subgroup of the elliptic curve E consisting of the points P such that n * P = 0.

Returns the univariate polynomial used to define the subgroup.

Returns the elliptic curve of which E is a subgroup.

Returns the set of rational points of E over its coefficient ring.