{
  "id": 4642,
  "curve_key": "258680289398231658230831455880327764247761:-120766067635315858812493396178945022035614330655621171823417241",
  "ainvs": [
    "1",
    "0",
    "0",
    "-5389172695796492879808988664173495088495",
    "139775541244578540291788629334314438134569541055324861316025"
  ],
  "rank_lower_bound": 12,
  "torsion": [
    2,
    2
  ],
  "family": null,
  "families": [],
  "naive_height": 286.0692345662864,
  "faltings_height": 21.803050924037954,
  "conductor": "232655514909455128819343175296085744158021265811427070",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "17",
    "41",
    "43",
    "47",
    "59",
    "61",
    "97",
    "109",
    "181",
    "251",
    "311",
    "331",
    "349",
    "461",
    "809",
    "971",
    "3347",
    "4621",
    "5849",
    "35141"
  ],
  "discriminant": "1577127184921640023068235036024080067291708302061776211382714398239870955501019243097386499486331320478447601591629440000",
  "regulator": "424887355739.29037164814025871614918311428362091223075252575306",
  "points": [
    [
      "664589994607916890",
      "369045044035967441916035800705"
    ],
    [
      "15799930667961736732",
      "242014971366598387635238147831"
    ],
    [
      "52832493462030148060",
      "50219408818337903104175452855"
    ],
    [
      "12692726059081896760",
      "270955926124550613629030249635"
    ],
    [
      "54675210776482909510",
      "92554293513238852175268721885"
    ],
    [
      "-50791491800014834874",
      "-531478135746980346031650880163"
    ],
    [
      "9857969417203597510",
      "295985193486239079363212648605"
    ],
    [
      "-29634988970924006528",
      "-522931355023613456936377786859"
    ],
    [
      "70879298019141692230",
      "337466278335892123131338586205"
    ],
    [
      "31944600286439789230",
      "14787837858166493367196911205"
    ],
    [
      "-76126424263489277858",
      "-329944938565453669610599577579"
    ],
    [
      "2213395861675027686730",
      "104076443455236884054187957598705"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "From Andrej Dujella and Juan Carlos Peral’s prescribed-torsion rank table, z2z2old1011121314.html: https://dujella.github.io/tors/z2z2old1011121314.html\r\n\r\nTwelve supplied rational points checked exactly on the displayed model; the source lists them as independent. Rational torsion Z/2Z × Z/2Z.",
  "created_at": "2026-10-19 16:13:00",
  "updated_at": "2026-10-19 16:13:00"
}