ar X iv : m at h / 04 03 11 6 v 1 [ m at h . N T ] 6 M ar 2 00 4 Elliptic Curves x 3 + y 3 = k of High Rank
نویسنده
چکیده
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 : x 3 + y = 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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