{
  "id": 510,
  "curve_key": "4297781290377621460836438658424214286759964722954179001:-8907546206991812316593724433452818191416046539162768710632614728494491477399512749",
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    "1",
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    "10309659961796079070131625494220076685011127717486239262727191317539981936305838"
  ],
  "rank_lower_bound": 17,
  "torsion": [
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  "naive_height": 377.3930817966992,
  "faltings_height": 29.174532209297162,
  "conductor": "7007913908914431737986612553664370128128260957329061089431076964775290140500425316602770",
  "bad_primes": [
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    "3",
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    "7",
    "11",
    "13",
    "17",
    "67",
    "83",
    "239",
    "307",
    "313",
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    "1231",
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    "11969",
    "14387",
    "61001",
    "93811",
    "112663",
    "513173",
    "580417"
  ],
  "discriminant": "22923749481637810847522061234251954659371561228027020443706000449553505366966302593000215673546427984304191609969320877843951964763047225162260896690519312500",
  "regulator": "34124710194134030283263937.7129188174111265903532983351012941909260500657",
  "points": [
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      "16404935786248860179653841335624471801237465102591926787410/82955501983139165171"
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      "9149454148003734166746454214487086749201558713227073169162635/242317665540886150668608"
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      "11151557313809957451242549051909482030130171741/96572036254413485089",
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      "1033116817465160999137765666557864201655699125374568832476151402114104/599559069643215922139320070047"
    ],
    [
      "542330106871978367862959904992419876210684837327331/1761994452186272607814564",
      "8071679069499505283340547275381374060108064477081413542047736606466468763645/2338873886339938593289171752567399512"
    ],
    [
      "-109594723283065934287890397180541738469373959595762327/1142583433484351565551782144",
      "163928407841925265080798729913297034345706415983530738891057683849524500154938541/38621744305904412386841402651490901577728"
    ],
    [
      "284154264372503388465413314494787277031230261/2015547309165075625",
      "2000393603008215230396006043179445193980307964606907326947476774066/2861471960546147844416703125"
    ],
    [
      "95867407361888906344024224369984644367000758217297374/578068332396972376634812201",
      "2056827915468524192195387359874713565977549744263532338852730989041878101434055/13898526770073301311542032169400533627051"
    ]
  ],
  "submitter": "Matthias Breddin",
  "history": [],
  "commentary": "Exact rank 17, unconditional: rational 2-torsion point, 17 independent points (Neron-Tate height matrix of rank 17; the archived certificate additionally saturates the lattice at all primes up to 97 — the points listed here are the ellrank output, not the saturated basis) with matching 2-descent upper bound 17 (PARI/GP 2.15.4 ellrank, effort 2). Fiber t=24693/1657 on branch u=29/13 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found by a Mestre-Nagao residue-table sieve followed by an exact 2-descent-type upper bound and ellrank (Matthias Breddin, 2026-09-02; method: doi:10.5281/zenodo.22242107).",
  "created_at": "2026-09-02 13:41:41",
  "updated_at": "2026-09-02 13:41:41"
}