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    MacLeod, Allan 2015. On two of John Leech's unsolved problems concerning Rational cuboids. Glasnik Matematicki, Vol. 50, Issue. 2, p. 269.

    Yokoyama, Shun’ichi 2012. On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields. American Journal of Computational Mathematics, Vol. 02, Issue. 04, p. 358.

    Cremona, J.E. and Fisher, T.A. 2009. On the equivalence of binary quartics. Journal of Symbolic Computation, Vol. 44, Issue. 6, p. 673.

    O’Neil, Catherine 2005. Models of some genus one curves with applications to descent. Journal of Number Theory, Vol. 112, Issue. 2, p. 369.

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  • LMS Journal of Computation and Mathematics, Volume 5
  • January 2002, pp. 7-17

Computing the Rank of Elliptic Curves over Number Fields

  • Denis Simon (a1)
  • DOI: http://dx.doi.org/10.1112/S1461157000000668
  • Published online: 01 February 2010
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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COPYRIGHT: © London Mathematical Society 2002
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6H. Cohen , Advanced topics in computational algebraic number theory, Grad. Texts in Math. 193 (Springer, 2000).
9J. E. Cremona , ‘Classical invariants and 2-descent on elliptic curves’, J. Symb. Comput. 31 (2001) 71–87.
10J. E. Cremona , D. Rusin , ‘Efficient solution of rational conics’, Math. Comp. (2002), to appear.
11J. E. Cremona , P. Serf , ‘Computing the rank of elliptic curves over real quadratic fields of class number 1‘, Math.Comp. 68 (1999) 1187–1200.
12J. E. Cremona and E. Whitley , ‘Periods of cusp forms and elliptic curves over imaginary quadratic fields’, Math.Comp. 62 (1994) 407–429.
14Z. Djabri , E. F. Schaefer and N. P. Smart , ‘Computing the P-Selmer group of an elliptic curve’, Trans. Amer. Math. Soc. 352 (2000) 5583–5597.
15S. Lang , Algebraic number theory, Grad. Texts in Math. 110, 2nd edn (Springer,1994).
18E. F. Schaefer , ‘2-descent on the Jacobians of hyperelliptic curves’, J.Number Theory 51 (1995) 219–232.
19E. F. Schaefer , ‘Class groups and Selmer groups‘, J.Number Theory 56 (1996) 79–114.
22D. Simon , ‘Solving norm equations using S-units’, Math.Comp. (2002), to appear.
25M. Stoll , ‘Implementing 2-descent for Jacobians of elliptic curves’, Acta Arith. 98 (2001) 245–277.
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