{
  "id": 2081,
  "curve_key": "1940830683700523088961:-84901934412436375007741608403041",
  "ainvs": [
    "1",
    "0",
    "0",
    "-40433972577094231020",
    "98266127788802232618276416400"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    4
  ],
  "naive_height": 147.0522090653358,
  "faltings_height": 10.104420186958443,
  "conductor": "3397916818895170749870",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "17",
    "19",
    "31",
    "401",
    "449",
    "641",
    "1272881"
  ],
  "discriminant": "59275829358511193519678529705531832109289309962649600000000",
  "regulator": "1423234.18739605823669318716674039928493446385584448045",
  "points": [
    [
      "-362579400",
      "335974863716820"
    ],
    [
      "3306984840",
      "26780271807780"
    ],
    [
      "8132750190",
      "554378516542530"
    ],
    [
      "37028359395/4",
      "5753512729540395/8"
    ],
    [
      "-306164821800/49",
      "112184200634108940/343"
    ],
    [
      "68175087165/16",
      "3698267202717795/64"
    ],
    [
      "564805090008/169",
      "47228177252075364/2197"
    ],
    [
      "607102583800/81",
      "339007592956732820/729"
    ]
  ],
  "submitter": "Seojun Jang",
  "history": [],
  "commentary": "Specialization of a known (\\mathbb{Z}/4\\mathbb{Z})-torsion family. Exact rank 8 verified by PARI/GP, with matching 2-descent upper bound.",
  "created_at": "2026-09-26 05:35:15",
  "updated_at": "2026-09-26 05:35:15"
}