{
  "id": 5566,
  "curve_key": "-112590451344:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "2345634403",
    "0"
  ],
  "rank_lower_bound": 7,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 76.34106824247927,
  "faltings_height": 4.429996098514763,
  "conductor": "176064024081189325088",
  "bad_primes": [
    "2",
    "7",
    "61",
    "101",
    "137",
    "397"
  ],
  "discriminant": "-825963664030916292165527604928",
  "regulator": "326170.4244321898549036263623275668041718748131734473",
  "points": [
    [
      "6161",
      "3832142"
    ],
    [
      "15372",
      "6299958"
    ],
    [
      "25011",
      "8620458"
    ],
    [
      "40097",
      "12590458"
    ],
    [
      "1339072/9",
      "1629578888/27"
    ],
    [
      "14475017/16",
      "55150576613/64"
    ],
    [
      "17121419/361",
      "101256071966/6859"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Found in my own run in the quartic-twist family y^2 = x^3 + A*x.\r\n\r\nThe construction A = u*(v^2-u), with (u,v) = (6161,622), gives A = 2345634403 and the rational point (6161,3832142). A point search seeded with this point produced the seven submitted points. The resulting model [0,0,0,2345634403,0] is already globally minimal.\r\n\r\nThe computation returned rank bounds [7,7], torsion Z/2Z, conductor 176064024081189325088, and minimal discriminant -825963664030916292165527604928. Exact quadratic-character certification at 13 good primes gives an F2 matrix of rank 8, with torsion-image rank 1, proving the seven submitted points independent modulo torsion. Together with the upper bound, this establishes rank exactly 7.",
  "created_at": "2026-10-23 11:15:53",
  "updated_at": "2026-10-23 11:15:53"
}