{
  "id": 6270,
  "curve_key": "11574816:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-241142",
    "0"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 48.79302678444579,
  "faltings_height": 2.134325977011973,
  "conductor": "3721565706496",
  "bad_primes": [
    "2",
    "11",
    "97",
    "113"
  ],
  "discriminant": "897425797595858432",
  "regulator": "1227.20211403219162544585333082499748513450208965",
  "points": [
    [
      "-352",
      "6424"
    ],
    [
      "-226",
      "6554"
    ],
    [
      "-88",
      "4532"
    ],
    [
      "2825/4",
      "107915/8"
    ],
    [
      "5537/4",
      "385217/8"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Quartic twist y^2 = x^3 + ax (j = 1728) of rank 5. Rank proven (independent generators plus a matching 2-descent upper bound) in M. Rubinstein's elliptic-curves-by-rank collection, file families/family_Z2.add_2026-10-24.part1.tsv.gz:\r\nhttps://github.com/mrubinst/elliptic-curves-by-rank",
  "created_at": "2026-10-31 23:31:29",
  "updated_at": "2026-10-31 23:31:29"
}