{
  "id": "math/0403116",
  "version": "v1",
  "published": "2004-03-06T16:56:49.000Z",
  "updated": "2004-03-06T16:56:49.000Z",
  "title": "Elliptic Curves x^3 + y^3 = k of High Rank",
  "authors": [
    "Noam D. Elkies",
    "Nicholas F. Rogers"
  ],
  "comment": "10 pages; to appear in the Proceedings of ANTS-VI (Algorithmic Number Theory Symposium, 2004)",
  "categories": [
    "math.NT"
  ],
  "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 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-03-06T16:56:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05"
    ],
    "keywords": [
      "elliptic curves",
      "high rank",
      "cubic surfaces",
      "relevant algorithms",
      "first examples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......3116E"
    }
  }
}