{
  "id": 6304,
  "curve_key": "6516518104128:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-135760793836",
    "0"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 88.51608394833431,
  "faltings_height": 5.44458074066935,
  "conductor": "36861986285961791189792",
  "bad_primes": [
    "2",
    "29",
    "53",
    "59",
    "241",
    "1553"
  ],
  "discriminant": "160141200657725378390100251511107584",
  "regulator": "38972935.4845099109027135339181055820201672440352620089",
  "points": [
    [
      "-355475",
      "57800235"
    ],
    [
      "-112572",
      "117712788"
    ],
    [
      "1795268",
      "2354229972"
    ],
    [
      "19427733",
      "85616026743"
    ],
    [
      "117715968",
      "1277176561728"
    ],
    [
      "-108224406407/304704",
      "9825968581184717/168196608"
    ],
    [
      "-11407286619/121801",
      "4635813546476865/42508549"
    ],
    [
      "-640381103/1764",
      "2813194185479/74088"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Quartic twist y^2 = x^3 + ax (j = 1728) of rank 8. 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 23:55:44",
  "updated_at": "2026-10-31 23:55:44"
}