{
  "id": 6311,
  "curve_key": "185321441226336:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-3860863358882",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 98.55933886022203,
  "faltings_height": 6.281518649993326,
  "conductor": "954001016061286344210555136",
  "bad_primes": [
    "2",
    "17",
    "19",
    "43",
    "257",
    "433",
    "1249"
  ],
  "discriminant": "3683267567247218824874362317014816317952",
  "regulator": "1714316666.1586061676287335585208207862160578489548598530",
  "points": [
    [
      "-1451808",
      "1595364888"
    ],
    [
      "-6664",
      "160401052"
    ],
    [
      "-46878693040433/24295041",
      "61719632165936983193/119750257089"
    ],
    [
      "-15443453435528/10055241",
      "48428253073353147868/31885169211"
    ],
    [
      "-187785182825/173889",
      "123695489537180405/72511713"
    ],
    [
      "17347361/4",
      "64410751393/8"
    ],
    [
      "34028050/9",
      "169587051130/27"
    ],
    [
      "8916543554/3721",
      "481930262550146/226981"
    ],
    [
      "108698352002/54289",
      "6886007222429842/12649337"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Quartic twist y^2 = x^3 + ax (j = 1728) of rank 9. 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 23:59:54",
  "updated_at": "2026-10-31 23:59:54"
}