{
  "id": 6276,
  "curve_key": "-568689024:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "11847688",
    "0"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 60.476532935797565,
  "faltings_height": 3.1079514896246208,
  "conductor": "334010020096",
  "bad_primes": [
    "2",
    "41",
    "881"
  ],
  "discriminant": "-106434102051495728939008",
  "regulator": "3354.66495667522907763892805305588055141265460783",
  "points": [
    [
      "200",
      "48760"
    ],
    [
      "4961",
      "425293"
    ],
    [
      "11849",
      "1343119"
    ],
    [
      "2009/4",
      "623651/8"
    ],
    [
      "16072/9",
      "4424392/27"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Quartic twist y^2 = x^3 + ax (j = 1728) of rank 5. 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 23:36:20",
  "updated_at": "2026-10-31 23:36:20"
}