Elliptic Curves x3 + y3 = k of High Rank

  • Noam D. Elkies
  • Nicholas F. Rogers
Conference paper

DOI: 10.1007/978-3-540-24847-7_13

Part of the Lecture Notes in Computer Science book series (LNCS, volume 3076)
Cite this paper as:
Elkies N.D., Rogers N.F. (2004) Elliptic Curves x3 + y3 = k of High Rank. In: Buell D. (eds) Algorithmic Number Theory. ANTS 2004. Lecture Notes in Computer Science, vol 3076. Springer, Berlin, Heidelberg

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 Ek: x3 + y3 = 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 Ek of a given rank, in the sense of both |k| and the conductor of Ek, 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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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Noam D. Elkies
    • 1
  • Nicholas F. Rogers
    • 2
  1. 1.Department of Mathematics, Supported in part by NSF grant DMS-0200687Harvard UniversityCambridgeUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

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