Elliptic Curves x^3 + y^3 = k of High Rank
| dc.creator | Elkies, Noam D. | |
| dc.creator | Rogers, Nicholas F. | |
| dc.date | 2004-03-06 | |
| dc.date.accessioned | 2026-07-07T05:06:09Z | |
| dc.date.available | 2026-07-07T05:06:09Z | |
| dc.description | 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 xy(x+y)=k 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. | |
| dc.description | 10 pages; to appear in the Proceedings of ANTS-VI (Algorithmic Number Theory Symposium, 2004) | |
| dc.identifier | https://arxiv.org/abs/math/0403116 | |
| dc.identifier | http://arxiv.org/abs/math/0403116 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70376 | |
| dc.subject | Number Theory | |
| dc.subject | 11G05 | |
| dc.title | Elliptic Curves x^3 + y^3 = k of High Rank | |
| dc.type | text |