Algorithms for Modular Elliptic Curves

Elliptic curves are of central and growing importance in computational number theory, with numerous applications in such areas as cryptography, primality testing and factorisation.

This book presents a thorough treatment of many algorithms concerning the arithmetic of elliptic curves, with remarks on computer implementation. An extensive set of tables is provided giving the results of the author's implementation of the algorithms.

Although the idea of using modular symbols for computing the modular elliptic curves defined over Q with conductor N is not new, neither the complete description of the algorithm nor the description of its implementation had been available before the writing of this book; moreover, the complete list of all the modular curves defined over Q with conductor less than 999...will prove very useful for any mathematician interested in the arithmetic of elliptic curves.