# Algorithms for Modular Elliptic Curves

## Contents of Second Edition

1. Introduction
2. Modular symbol algorithms
1. Modular Symbols and Homology
1. The upper half-plane, the modular group and cusp forms
2. The duality between cusp forms and homology
3. Real structure
4. Modular symbol formalism
5. Rational structure and the Manin-Drinfeld Theorem
6. Triangulations and homology
2. M-symbols and \$\Gamma_0(N)\$
3. Conversion between modular symbols and M-symbols
4. Action of Hecke and other operators
5. Working in \$H^+(N)\$
6. Modular forms and modular elliptic curves
7. Splitting off one-dimensional eigenspaces
8. \$L(f,s)\$ and the evaluation of \$L(f,1)/\period(f)\$
9. Computing Fourier coefficients
10. Computing periods I
11. Computing periods II: Indirect method
12. Computing periods III: Evaluation of the sums
13. Computing \$L^{(r)}(f,1)\$
14. Obtaining equations for the curves
15. Computing the degree of a modular parametrization
1. Modular Parametrizations
2. Coset representatives and Fundamental Domains
3. Implementation for \$\Gamma_0(N)\$
Appendix to Chapter II. Examples
1. Example 1. N=11
2. Example 2. N=33
3. Example 3. N=37
4. Example 4. N=49
3. Elliptic curve algorithms
1. Terminology and notation
2. The Kraus--Laska--Connell algorithm and Tate's algorithm
3. The Mordell--Weil group I: finding torsion points
4. Heights and the height pairing
5. The Mordell--Weil group II: generators
6. The Mordell--Weil group III: the rank
7. The period lattice
8. Finding isogenous curves
9. Twists and complex multiplication
4. The tables
1. Elliptic curves
2. Mordell--Weil generators
3. Hecke eigenvalues
4. Birch--Swinnerton-Dyer data
5. Parametrization degrees
5. Bibliography

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