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

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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.