{
  "id": 5609,
  "curve_key": "-6931247:-9884515609",
  "ainvs": [
    "1",
    "0",
    "0",
    "144401",
    "11452445"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    4
  ],
  "family": null,
  "families": [],
  "naive_height": 47.25465089177507,
  "faltings_height": 2.0263005171712756,
  "conductor": "9972582",
  "bad_primes": [
    "2",
    "3",
    "1033",
    "1609"
  ],
  "discriminant": "-249245312098403268",
  "regulator": "338.48271946285053895507774560386370197975640",
  "points": [
    [
      "3059/4",
      "187819/8"
    ],
    [
      "2086",
      "95863"
    ],
    [
      "-266/9",
      "72641/27"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Obtained in my own run using a Tate-normal-form discriminant sieve, following the method reported by jesper-petersen for ICARM curve #4087:\r\nhttps://elliptic-rank.icarm.cloud/curve/4087\r\n\r\nSpecialization at b = u/v = -1609/9216, with (u,v) = (-1609,9216), of the order-4 Tate family y^2 + xy - by = x^3 - bx^2. Clearing denominators gives [9216,14828544,136659861504,0,0], then reduction gives the global minimal model [1,0,0,144401,11452445].\r\n\r\nThree independent rational points were obtained on this specialization and transported to the minimal model. All three were checked exactly. Matching rank bounds [3,3] establish rank exactly 3, with rational torsion Z/4Z and conductor 9972582.",
  "created_at": "2026-10-23 22:36:21",
  "updated_at": "2026-10-23 22:36:21"
}