{
  "id": 6318,
  "curve_key": "1318379233150080:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-27466234023960",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 104.44553857021454,
  "faltings_height": 6.772035292492701,
  "conductor": "335286227315083526157369600",
  "bad_primes": [
    "2",
    "3",
    "5",
    "31",
    "173",
    "1733",
    "8209"
  ],
  "discriminant": "1326103197760329676625208318453644488704000",
  "regulator": "1270802461.7369226721204106991760354459245493640571410049",
  "points": [
    [
      "-3475224",
      "7313029704"
    ],
    [
      "223057896",
      "3330477445176"
    ],
    [
      "541233960",
      "12590912243880"
    ],
    [
      "-686655850599/170569",
      "474287615781591879/70444997"
    ],
    [
      "-9612951/16",
      "258270707403/64"
    ],
    [
      "767511040/9",
      "21222927461440/27"
    ],
    [
      "5576669224/9",
      "416434587634072/27"
    ],
    [
      "69867463929/7921",
      "14854403035027593/704969"
    ],
    [
      "18174680139731944/1577440089",
      "2182040102961846219543688/62651188014813"
    ]
  ],
  "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": "2026-11-01 00:04:26",
  "updated_at": "2026-11-01 00:04:26"
}