This chapter contains descriptions for functions pertaining to arithmetic with elliptic curves. The category to which elliptic curves belong is called CurveEll; points on curves are of type CurveEllPt; and there also exists a special category for Kodaira symbols KodSym.

This module is currently under construction; the main functions that are present deal with elementary arithmetic over fields, and with more sophisticated questions over the rational field. The latter include functions for minimal models, local information (Tate's algorithm), and the computation of the Mordell-Weil group. These facilities are based on implementations by John Cremona. We refer to his book [J.E. Cremona, Algorithms for modular elliptic curves, Cambridge: Cambridge University Press, 1992.] for details.

Some functions for computing with elliptic curves over finite fields have also been implemented. Many of the algorithms, as well as background material, can be found in Menezes, A., Elliptic curve public key cryptosystems (Kluwer Academic Publishers, 1993), and Connell, I., The elliptic curve handbook (published on the Internet).

For the time being, there is a restriction on elliptic curves to be defined over fields.