| A widely believed conjecture states that the ranks of the elliptic curves Ek : x3 + y 3 = k can be arbitrarily large. In this work we apply the observation that pairs of rational points on Ek correspond to rational points on cubic surfaces to construct the first known examples of rank 8, 9, 10, and 11. As a corollary we produce examples of elliptic curves over Q with a rational 3-torsion point and rank as high as 11, which are the first known examples of elliptic curves with nontrivial 3-torsion and rank larger than 8. 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. Many of our techniques are applicable to other families of twists of elliptic curves, notably the so-called "congruent number elliptic curves" Ed : y 2 = x3 - d 2x. We include an explicit formula for descent via 3-isogeny for the curves Ek, descriptions of the relevant algorithms and heuristics, as well as numerical data. |
