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

摘要

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~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.