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Computing the Rank of Elliptic Curves over Number Fields

  • Denis Simon (a1)
Abstract
Abstract

This paper describes an algorithm of 2-descent for computing the rank of an elliptic curve without 2-torsion, defined over a general number field. This allows one, in practice, to deal with fields of degree from 1 to 5.

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6 H. Cohen , Advanced topics in computational algebraic number theory, Grad. Texts in Math. 193 (Springer, 2000).

9 J. E. Cremona , ‘Classical invariants and 2-descent on elliptic curves’, J. Symb. Comput. 31 (2001) 7187.

10 J. E. Cremona , D. Rusin , ‘Efficient solution of rational conics’, Math. Comp. (2002), to appear.

11 J. E. Cremona , P. Serf , ‘Computing the rank of elliptic curves over real quadratic fields of class number 1‘, Math.Comp. 68 (1999) 11871200.

12 J. E. Cremona and E. Whitley , ‘Periods of cusp forms and elliptic curves over imaginary quadratic fields’, Math.Comp. 62 (1994) 407429.

14 Z. Djabri , E. F. Schaefer and N. P. Smart , ‘Computing the P-Selmer group of an elliptic curve’, Trans. Amer. Math. Soc. 352 (2000) 55835597.

15 S. Lang , Algebraic number theory, Grad. Texts in Math. 110, 2nd edn (Springer,1994).

18 E. F. Schaefer , ‘2-descent on the Jacobians of hyperelliptic curves’, J.Number Theory 51 (1995) 219232.

19 E. F. Schaefer , ‘Class groups and Selmer groups‘, J.Number Theory 56 (1996) 79114.

22 D. Simon , ‘Solving norm equations using S-units’, Math.Comp. (2002), to appear.

25 M. Stoll , ‘Implementing 2-descent for Jacobians of elliptic curves’, Acta Arith. 98 (2001) 245277.

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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
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