Elliptic Curves x3 + y3 = k of High Rank
- Cite this paper as:
- Elkies N.D., Rogers N.F. (2004) Elliptic Curves x3 + y3 = k of High Rank. In: Buell D. (eds) Algorithmic Number Theory. ANTS 2004. Lecture Notes in Computer Science, vol 3076. Springer, Berlin, Heidelberg
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: x3 + y3 = 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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