{
  "id": 486,
  "curve_key": "32780217342456787200259118896446281096203397346282081:-5894169505059787505205466438128238202796931704711500995447120562496705249973521",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-682921194634516400005398310342630856170904111380877",
    "6821955445671050353247067807452796578532956141061962997054844323393510773429"
  ],
  "rank_lower_bound": 16,
  "torsion": [
    2
  ],
  "naive_height": 362.76499483898203,
  "faltings_height": 28.07952978733415,
  "conductor": "6467420206652532321375335216913420856551465130881758563694664862777597598401736912029710",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "19",
    "31",
    "43",
    "61",
    "89",
    "131",
    "199",
    "211",
    "263",
    "283",
    "337",
    "443",
    "619",
    "809",
    "1873",
    "3083",
    "60317",
    "738937",
    "3584363",
    "7317619",
    "32849761151039335443737"
  ],
  "discriminant": "279228833446562741971527345897062313521266003657925183217334855335599415572366692915757669651945351089113804714819908088282239548453240121026560000000000",
  "regulator": "7172760148631977039.636466213996417733839847213041880300767434487524233",
  "points": [
    [
      "-9762391056417557724227313",
      "112064720434715281659675947942120753656"
    ],
    [
      "10199426602031421175212687",
      "30291581125967387741227549677016913656"
    ],
    [
      "-26703193566724898198579163",
      "77570216542872641214250610195929927756"
    ],
    [
      "74339631351274555932687",
      "82287227117269256429427209646133073656"
    ],
    [
      "8143638333462863185930437",
      "42433111094730504144586715967253726156"
    ],
    [
      "6347306706614733462487633",
      "52373346074850275066976699893352771468"
    ],
    [
      "-28772647905236080099758063",
      "51493015757249490161724968609247766156"
    ],
    [
      "1301108132880556040430486243/169",
      "98756766959721788446292536288843732084012/2197"
    ],
    [
      "62105722728783024006723987",
      "451617001178015208216478957845570761656"
    ],
    [
      "13853847957299042912492637",
      "4452601827342210622364850072522422956"
    ],
    [
      "1201183118016690361530187",
      "77481450372659130192040955011523736156"
    ],
    [
      "34512803858690635684452687",
      "156082643341551822859487225491756073656"
    ],
    [
      "13557962909925774806526039",
      "7425476547694768209980691301019144848"
    ],
    [
      "75356667225628874837100427813/4624",
      "1431570522933894048835279774541524415662767/314432"
    ],
    [
      "29822020305213842678468937",
      "113921753006743501549783126243157344906"
    ],
    [
      "576001391543572697044040838807/32761",
      "93735370102407504555104965633533572570803096/5929741"
    ]
  ],
  "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=-4046/2511 on branch u=19/13 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 19:28:31",
  "updated_at": "2026-09-01 19:28:31"
}