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International Algorithmic Number Theory Symposium

ANTS 2004: Algorithmic Number Theory pp 184–193Cite as

Elliptic Curves x 3 + y 3 = k of High Rank

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3076))

Abstract

We use rational parametrizations of certain cubic surfaces and an explicit formula for descent via 3-isogeny to construct the first examples of elliptic curves E k : x 3 + y 3 = k of ranks 8, 9, 10, and 11 over ℚ. As a corollary we produce examples of elliptic curves over ℚ with a rational 3-torsion point and rank as high as 11. We also discuss the problem of finding the minimal curve E k of a given rank, in the sense of both |k| and the conductor of E k , and we give some new results in this direction. We include descriptions of the relevant algorithms and heuristics, as well as numerical data.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Supported in part by NSF grant DMS-0200687, Harvard University, Cambridge, MA, 02140, USA

    Noam D. Elkies

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02140, USA

    Nicholas F. Rogers

Authors
  1. Noam D. Elkies
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  2. Nicholas F. Rogers
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, University of South Carolina , SC 29208, Columbia, USA

    Duncan Buell

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© 2004 Springer-Verlag Berlin Heidelberg

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Elkies, N.D., Rogers, N.F. (2004). Elliptic Curves x 3 + y 3 = k of High Rank. In: Buell, D. (eds) Algorithmic Number Theory. ANTS 2004. Lecture Notes in Computer Science, vol 3076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24847-7_13

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  • DOI: https://doi.org/10.1007/978-3-540-24847-7_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22156-2

  • Online ISBN: 978-3-540-24847-7

  • eBook Packages: Springer Book Archive

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