##
**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.

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.

Reviewer: G.Pfister (Kaiserslautern)

### MSC:

14Q05 | Computational aspects of algebraic curves |

14H52 | Elliptic curves |

11Y16 | Number-theoretic algorithms; complexity |

68W30 | Symbolic computation and algebraic computation |

14G35 | Modular and Shimura varieties |

14-04 | Software, source code, etc. for problems pertaining to algebraic geometry |

### Citations:

Zbl 0758.14042
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\textit{J. E. Cremona}, Algorithms for modular elliptic curves. Cambridge: Cambridge University Press (1997; Zbl 0872.14041)

### Digital Library of Mathematical Functions:

§23.20(ii) Elliptic Curves ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions### Online Encyclopedia of Integer Sequences:

Glaisher’s chi numbers. a(n) = chi(4*n + 1).Number of solutions of the congruence y^2 == x^3 - 4*x^2 + 16 (mod p) as p runs through the primes.

The p-defect p - N(p) of the congruence y^2 == x^3 + 4*x (mod p) for primes p, where N(p) is the number of solutions given by A276730.