{
  "id": 489,
  "curve_key": "25037115971601301747969502955240491884423826615881825:-3945164941772372226416894958998447898048661502639595883284944474971559632260625",
  "ainvs": [
    "1",
    "0",
    "0",
    "-521606582741693786416031311567510247592163054497538",
    "4566163127051356743538072825743951320192950093385811863916905842981272810692"
  ],
  "rank_lower_bound": 16,
  "torsion": [
    2
  ],
  "naive_height": 361.95659731632566,
  "faltings_height": 27.991392197636415,
  "conductor": "13742738586193407312966995954537545897263198361682836147475206656230039988267492498287005950",
  "bad_primes": [
    "2",
    "3",
    "5",
    "17",
    "19",
    "23",
    "29",
    "43",
    "167",
    "269",
    "397",
    "499",
    "661",
    "701",
    "733",
    "853",
    "1399",
    "1699",
    "5807",
    "7001",
    "13381",
    "32887",
    "100787",
    "101089",
    "194863",
    "40236799",
    "53264824471"
  ],
  "discriminant": "75445254481781595267746905315098659307321544309204814184568588269404249343994019608220476659259323537058499274987211126964564045512340886932530176000000",
  "regulator": "24344958461571653650.09786581344328485485175124500312205662062824277496",
  "points": [
    [
      "-17219631410126251154144668",
      "91881146379649528412098378937192884334"
    ],
    [
      "8984833943268959740083578",
      "24595435600483527448824758957291636612"
    ],
    [
      "8558463623378774295412772",
      "26998083487837782855289535816046852014"
    ],
    [
      "17162864741912187603179132",
      "25873939037360593175299474031314422134"
    ],
    [
      "27239229787567201070738372",
      "102804895311338736914281503410138601614"
    ],
    [
      "-26188498992214911046038148",
      "16285037725366346492747633281486061174"
    ],
    [
      "12452827844565546507592532",
      "1335118799301180009064346649153334334"
    ],
    [
      "29716951865008537640344652",
      "123727778757298476488726777887178592974"
    ],
    [
      "115824909915473128940098376",
      "1223923916853011780477540012245429585994"
    ],
    [
      "1165123299182324293094557132/81",
      "4578179678176510757981597713779120136606/729"
    ],
    [
      "7854011508927619753250966",
      "30885880669459248216865223435832159080"
    ],
    [
      "-1137354004910603596212884105788/43681",
      "203112635868112617474730918150756212315242446/9129329"
    ],
    [
      "62529958279512719520792163286/6241",
      "9169052650586973837917455100753393827824920/493039"
    ],
    [
      "886366473263900732534372",
      "64066566239564254359849993708666381614"
    ],
    [
      "176016648310258585729161929081332/8696601",
      "1229981036185603252206440254761801730392495109766/25646276349"
    ],
    [
      "73055204321308695045930087991892/175561",
      "623498543912818758241380349853896745479422689626/73560059"
    ]
  ],
  "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=-4182/133 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 19:28:43",
  "updated_at": "2026-09-02 05:25:52"
}