Subsections

• Creation of Subschemes of Elliptic Curves

A subscheme of an elliptic curve E is defined by a polynomial psi is the following way. From an algebro-geometric perspective, an elliptic curve E defined by a Weierstrass equation y^2 = f(x) over a field K is the projective variety (embedded in the projective plane over K) corresponding to the ideal of k[x, y] generated by y^2 - f(x). The subscheme E_( psi ) of E defined by a polynomial psi is the variety corresponding to the ideal generated by y^2 - f(x) and psi. Note that the set of rational points of E_( psi ) may not form a subgroup of the rational points of E.

Subscheme(E, I) : CurveEll, RngMPol -> CurveEll

Subscheme(E, r) : CurveEll, RngMPolElt -> CurveEll

Creates the subscheme of the elliptic curve E defined by the ideal I. If the function is passed a multivariate polynomial, it creates the subscheme defined by the principal ideal generated by that polynomial.

Returns true iff the subscheme Es of an elliptic curve E defines a subgroup of the rational points of E.

Returns the univariate polynomial used to define the subscheme.

Returns the elliptic curve of which E is a subscheme.

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