{
  "id": 3598,
  "curve_key": "487761197722904011955988124861209:-10769137512333387061262759407696645264550447153277",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-10161691619227166915749752601275",
    "12464279528163644972439987454588772067704871561"
  ],
  "rank_lower_bound": 11,
  "torsion": [
    4
  ],
  "naive_height": 225.80213618092733,
  "faltings_height": 16.548012740308934,
  "conductor": "3482515236760862602092865033854708270",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "17",
    "19",
    "31",
    "59",
    "79",
    "113",
    "173",
    "181",
    "251",
    "1231",
    "330217",
    "25236637"
  ],
  "discriminant": "40176345626093609450376048624803082963114812350662412021098091913921373931715169100311518700",
  "regulator": "5645080120695.40741510016139005863307539396397797285344993750",
  "points": [
    [
      "1020468590725723",
      "56189532674193978957666"
    ],
    [
      "-825906706757175",
      "-142455325026766108707894"
    ],
    [
      "909643946299815",
      "63035267962225695575511"
    ],
    [
      "3915571700257729653/1936",
      "1159628794065718672206509169/85184"
    ],
    [
      "3939133353161269188/529",
      "7194876276593453311803239025/12167"
    ],
    [
      "380898756026071861183/11236",
      "7402114018410477358552707047591/1191016"
    ],
    [
      "49232485319772499/25",
      "1186416245486369941688583/125"
    ],
    [
      "470677769767844227/324",
      "161625538160579580674756237/5832"
    ],
    [
      "-1072940074906780955/729",
      "-3063982516357210035768418418/19683"
    ],
    [
      "498179707901428298100/223729",
      "3135133428571551444517184488527/105823817"
    ],
    [
      "-52827570631332100607208/22118209",
      "-15813320896239343788958108361228253/104021936927"
    ]
  ],
  "submitter": "dujella",
  "history": [],
  "commentary": "It is the fiber at t = 35033/156392 of Elkies' Z/4Z fibration E1 of the singular K3 surface of discriminant −163. The fiber also lies on the rank-5 subfamily, found by Dujella-Peral, defined by the condition 199664t² + 104728t − 122161 = □, parametrized by t = (−u² − 330u + 94936)/(u² + 199664), at u = 7052/65. The curve was found with the assistance of Claude, based on our old PARI/GP program.",
  "created_at": "2026-09-29 08:36:14",
  "updated_at": "2026-09-29 08:36:14"
}