{
  "id": 490,
  "curve_key": "327417174994913800963614900080195491111529441404627441:-186504732264743464166300688863281754785688869341264847538649801889805322681196105",
  "ainvs": [
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    "0",
    "0",
    "-6821191145727370853408643751670739398156863362596405",
    "215861958639749379822107278208513879969118249879447334974949728266795324738721"
  ],
  "rank_lower_bound": 16,
  "torsion": [
    2
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  "naive_height": 369.6692245950239,
  "faltings_height": 28.63736716547041,
  "conductor": "3752162734444495288724086395434552280850923039425413583723536398069902141272495003060217447262",
  "bad_primes": [
    "2",
    "3",
    "59",
    "67",
    "97",
    "103",
    "149",
    "157",
    "233",
    "349",
    "359",
    "521",
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    "3943",
    "4513",
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    "53791",
    "398423",
    "429389",
    "434267",
    "1038119",
    "3154201",
    "3622831",
    "18820031"
  ],
  "discriminant": "182733193787408683303404850123569749726008492456090345250402562272304830565755456514970303298675275064225076199599883001104915141243888605882861929772613632",
  "regulator": "212781497662551853783.3199860734366452299020061886728527888613025117248",
  "points": [
    [
      "-58066367208452794627045190",
      "645105575813064804351456766765542947359"
    ],
    [
      "108683139288816572978211034",
      "870793559991941262073246104563922624871"
    ],
    [
      "30949119552185734928872530",
      "185463340527999753493717028540989346199"
    ],
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      "101728736553943252869515218",
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    ],
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      "176948785149520450874910610524347373991"
    ],
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      "211493935084280025054400338",
      "2869368991194281157414919217632028559639"
    ],
    [
      "298422002992759285781301618",
      "4975585028913485900020447684384074270903"
    ],
    [
      "14922158072730348181991823904/289",
      "178237958308734550519170143408413123959359/4913"
    ],
    [
      "88330807598635715414633986",
      "550024440160711143669464793495515883271"
    ],
    [
      "3522490673748984497389209195344921266/47586368449",
      "3543734140026335294549902056703046765943268161902522665/10380633172570207"
    ],
    [
      "8722334662699927308263848019337730/260467321",
      "664547308597788961072466150920187491933833618872981/4203682093619"
    ],
    [
      "2417584218570072504032632954595233042978/86647446188521",
      "175343748081669353533522459403869103990295753088139615597443/806554373595446376181"
    ],
    [
      "25587556692707521861944308692473394/507826225",
      "106851636079941328203243615872303569551697264656977/11443863980375"
    ],
    [
      "955585254184374261426650705824652434/22493100529",
      "176910200462967277758193631337142573969165104198848719/3373447738037833"
    ],
    [
      "-52701156310256007821094445956221/562500",
      "76086575652733017393330989083048679632056493417/421875000"
    ]
  ],
  "submitter": "Matthias Breddin",
  "history": [],
  "commentary": "Exact rank 16, unconditional: rational 2-torsion point, 16 independent points with matching 2-descent upper bound (PARI/GP ellrank). Fiber t=-9956/75 on branch u=22/17 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found in an exhaustively certified slice search by Matthias Breddin. Preprint (proof, measurements, decided slices): https://doi.org/10.5281/zenodo.22242107 — data and code incl. the per-fiber ellrank certificate and witness points for this curve: https://doi.org/10.5281/zenodo.22242109 (Breddin, 2 Sep 2026).",
  "created_at": "2026-09-01 19:28:47",
  "updated_at": "2026-09-01 19:28:47"
}