{
  "id": 6297,
  "curve_key": "1238059941936:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-25792915457",
    "0"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 83.53370012215179,
  "faltings_height": 5.029382088487472,
  "conductor": "21288783608702384603168",
  "bad_primes": [
    "2",
    "17",
    "73",
    "113",
    "193",
    "953"
  ],
  "discriminant": "1098199591603255951087621436735552",
  "regulator": "105251399.045924234909355025985979879807789074549793291",
  "points": [
    [
      "-120409",
      "36877772"
    ],
    [
      "-3481",
      "9473276"
    ],
    [
      "-1921",
      "7038544"
    ],
    [
      "-484",
      "3533222"
    ],
    [
      "-27064969/169",
      "10609467848/2197"
    ],
    [
      "300931769/1681",
      "2306309566436/68921"
    ],
    [
      "1654073497/9216",
      "30031286495293/884736"
    ],
    [
      "109019021315481/632975281",
      "411205931938523619252/15925025094679"
    ]
  ],
  "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 23:51:28",
  "updated_at": "2026-10-31 23:51:28"
}