{
  "id": 6296,
  "curve_key": "-1040486050464:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "21676792718",
    "0"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 83.01212722836094,
  "faltings_height": 4.985917680671569,
  "conductor": "30072533922504820961536",
  "bad_primes": [
    "2",
    "7",
    "17",
    "31",
    "73",
    "167",
    "241"
  ],
  "discriminant": "-651876084343160479338917322894848",
  "regulator": "14281480.1510130416357812694954305516989294434650511960",
  "points": [
    [
      "8194",
      "13348026"
    ],
    [
      "8978",
      "13976334"
    ],
    [
      "1287904",
      "1471108344"
    ],
    [
      "4097/4",
      "37696497/8"
    ],
    [
      "321976/25",
      "2096546724/125"
    ],
    [
      "1778098/1089",
      "213810950206/35937"
    ],
    [
      "8001441/4",
      "22694753823/8"
    ],
    [
      "181721551/6561",
      "13250208216223/531441"
    ]
  ],
  "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 23:50:54",
  "updated_at": "2026-10-31 23:50:54"
}