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

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

2004-03-06

arxiv.org/abs/math/0403116v1

Mathematics

Number Theory

10 pages; to appear in the Proceedings of ANTS-VI (Algorithmic Number Theory Symposium, 2004)

Scientific paper

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 xy(x+y)=k 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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