{
  "id": 487,
  "curve_key": "779029826915714622693169537241757190979355943201:-1927733104521363945219773553965436036426694351991066243393514087115604785",
  "ainvs": [
    "1",
    "0",
    "0",
    "-16229788060744054639441032025869941478736582150",
    "2231172574677504566226588409051731091267083694718472662914036595001956"
  ],
  "rank_lower_bound": 16,
  "torsion": [
    2
  ],
  "naive_height": 332.8849429018246,
  "faltings_height": 25.793228151992874,
  "conductor": "1153464240454909671490191378607928470096426970591096070526119088776917088138",
  "bad_primes": [
    "2",
    "3",
    "7",
    "11",
    "13",
    "23",
    "31",
    "41",
    "43",
    "53",
    "61",
    "71",
    "73",
    "83",
    "101",
    "503",
    "2371",
    "4597",
    "4943",
    "33503",
    "55661",
    "175261",
    "385709",
    "1235239",
    "1657021",
    "5017709"
  ],
  "discriminant": "-1876951088311557689801031867173545551996530224234226596557815816398336270726531377514044949885697047169521540877941918663544361139837710532608",
  "regulator": "2485704261577274679.896116864024013223100454186381227638500724045938413",
  "points": [
    [
      "16734849667799038004684",
      "44319929974000082826896938646874890"
    ],
    [
      "-11540026265682030394384",
      "49162261384398890643734381027359046"
    ],
    [
      "237040688947921347490820",
      "108180233781379500625245716140578506"
    ],
    [
      "1329008925423939652145708",
      "1525792653688211206948273475931301418"
    ],
    [
      "312046144831912554815948",
      "165986552285268789436876089909462794"
    ],
    [
      "-12677641695845200071426508/169",
      "120865385457740551862780690677237008338/2197"
    ],
    [
      "446404395463549416141164",
      "289731256352779343849619321868673450"
    ],
    [
      "1178904038299649308852214",
      "1273403317635057139281402416392777550"
    ],
    [
      "232680361350911431110866",
      "105129298578224661799360170991900046"
    ],
    [
      "3103921437861916789052917052/52441",
      "461668347262127852240934271623975527597906/12008989"
    ],
    [
      "-143970089360107455970516",
      "39795136721842277254497573970504490"
    ],
    [
      "-103259420461270604948217220/2809",
      "7846973801082774387899550688501994872162/148877"
    ],
    [
      "138768535940101935698007438380/128881",
      "51376958746620981034928717392317765376502854/46268279"
    ],
    [
      "932659834078475926859945173557524/11098411801",
      "44686789127263894494013135478584502138177622171194/1169206584823549"
    ],
    [
      "-1031377801517774091392648/9",
      "1373051092654952213040309794293592426/27"
    ],
    [
      "1241047655482511415294226546388/6620329",
      "1294628634489126547884397533150710523876427858/17034106517"
    ]
  ],
  "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=-5259/12143 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 19:28:35",
  "updated_at": "2026-09-01 19:28:35"
}