Algorithms for modular elliptic curves. (English) Zbl 0872.14041

Cambridge: Cambridge University Press. 376 p. £45.00; $ 69.95 (1997).
[The first edition of this book appeared 1992; Zbl 0758.14042).]
This second edition of the book appears under the impression of the Wiles’ results. Now every elliptic curve defined over the rationals and of conductor less than 1000 is isomorphic to one of those in the table 1 of the book.
Chapter 2 is partially rewritten (2.1, 2.14). Section 2.15 is added. It is shown there how to compute the degree of the modular parametrization map for a modular elliptic curve.
The implementation changed from Algol 68 to \(C_{++}\). There are also improvements in chapter 3 concerning formulae for the Weierstraß coefficients, the global canonical height. The two-descent algorithms in section 3.6 are changed (theoretically and practically).
The two main changes in the tables are to include all the dates for \(N=702\) in tables 1-4 and include the new table 5 giving the degree of the modular parametrization for each strong Weil curve.


14Q05 Computational aspects of algebraic curves
14H52 Elliptic curves
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G05 Elliptic curves over global fields
11Y16 Number-theoretic algorithms; complexity
11Y35 Analytic computations
68W30 Symbolic computation and algebraic computation
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
11-02 Research exposition (monographs, survey articles) pertaining to number theory


Zbl 0758.14042


ecdata; ALGOL 68