Modular Curves (New) [HB 87]

A package for computing with modular curves has been developed by David Kohel. Modular curves in Magma are a special type of plane curve. A modular curve X is defined in terms of standard affine modular equations which are stored in precomputed databases. Those modular curves presently available are defined by modular equations--a bivariate polynomial relation between the j-invariant and one of several standard functions on X₀(N). These give singular, affine models for X₀(N)designed for computing isogenies of elliptic curves.

Features:

• Creation of a modular curve of specified level rom a database. Possible model types are Atkin, Canonical and Classical.
• Database of modular equations: Atkin, Canonical and Classical.
• Parametrization of the isogenies of an elliptic curve by points on some X₀(N).
• j-invariant of a modular curve over a field.
• Base curve of a modular curve.
• Automorphisms: Atkin-Lehner involution.
• Hilbert and Weber class polynomials.