Cremona, J. E. Algorithms for modular elliptic curves. (English) Zbl 0758.14042 Cambridge: Cambridge University Press. 343 p. (1992). Elliptic curves become more and more important in computational number theory. They are used in cryptography, primality testing, factorisation etc.The book under review contains many algorithms and remarks to their implementation concerning elliptic curves. It starts with an algorithm for the computation of modular elliptic curves using modular symbols. One can find algorithms to compute torsion and non-torsion points, to compute heights, to find isogenies and periods, to compute the rank etc.At the end of the book one can find a lot of tables with the results of these algorithms. Reviewer: Gerhard Pfister (Berlin) Cited in 7 ReviewsCited in 127 Documents MSC: 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 Keywords:torsion points; computation of modular elliptic curves; modular symbols; non-torsion points; heights; isogenies; periods; rank Software:ecdata × Cite Format Result Cite Review PDF 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.