{
  "id": 520,
  "curve_key": "3936226863640475752427242623486642110309319984241:-7327595043551308801969969701982888514708936698896926617041338082107877385",
  "ainvs": [
    "1",
    "0",
    "0",
    "-82004726325843244842234221322638377298110833005",
    "8481012781888088891168939580604853071753828770871517809118255345055441"
  ],
  "rank_lower_bound": 17,
  "torsion": [
    2
  ],
  "naive_height": 335.6829212380897,
  "faltings_height": 25.92336006211198,
  "conductor": "152600141322569268506676653888653486589666000303661800852107237946587629395343172143438",
  "bad_primes": [
    "2",
    "3",
    "11",
    "13",
    "17",
    "19",
    "29",
    "41",
    "83",
    "89",
    "109",
    "149",
    "277",
    "433",
    "563",
    "709",
    "827",
    "1091",
    "22109",
    "40531",
    "52711",
    "126001",
    "1354547",
    "2004787",
    "2350771",
    "21578390361659"
  ],
  "discriminant": "4220940463743691817552695733706774883678632857331130936686306134493474236265481798620611739985281683025401881296062354061307149651253900972032",
  "regulator": "50838263571662937892965497.8835273402068593338854693274000085039398589032",
  "points": [
    [
      "245476870488308107738680139714/1885129",
      "8429474408541374438936931176557872107850251/2588282117"
    ],
    [
      "-75000097556746912456526",
      "119203601498349699762418502026038655"
    ],
    [
      "-1139902671662317995457124515838/8048569",
      "2999349736911779194282491925977651423131995667/22833790253"
    ],
    [
      "31879702392084945903839262477364579/159912012100",
      "478165629889083709832067833481982154725453969231103/63947214518669000"
    ],
    [
      "-140332235587149272017976334535499836754301902/716530002437355646401",
      "2502943461589668954428348207010081448161908283155097108695026808111/19180131029693296176032813951649"
    ],
    [
      "1592363132855469348073634146606/7890481",
      "272121135006978790313553840612622102847797987/22164361129"
    ],
    [
      "5208127346571391435748013203075852470/139556786329",
      "11885286126730591397539646924947912815174651667545653527/52134647339283517"
    ],
    [
      "402566860049639694181285330/1849",
      "2447202809602741882475396125945981925677/79507"
    ],
    [
      "41413284326942771481253413432801235381468354204/203867031548089812784561",
      "1318797583504485669712576303790078707794192808190635645522388177473741/92049301859915190709770036814558441"
    ],
    [
      "484138088553303673277996005405939162/1219895329",
      "336863350396121122396778863299107903467475384037185569/42607284155983"
    ],
    [
      "505577127680788355907487186036594/223412809",
      "11280729557641509629558328147645275806601725182469/3339351256123"
    ],
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      "2619282797331396713941769057198386/10536817201",
      "63593089803395572753093360789579977780509596261447/1081593748865449"
    ],
    [
      "797401401912118680520112229056061604/13676838564841",
      "3157937960534410377018214503430055223584391013459084795/50579971594104847861"
    ],
    [
      "2571560684845442921658120523303030555179298/23873465263518016441",
      "3494697470976783657288805601846200827225540425410620571447658723/116646897705224447729393513261"
    ],
    [
      "12847158138409005694733772760796043749530/114239067281246209",
      "1007717508377293522799789167145040035518414560335322990563113/38611971138587267388412927"
    ],
    [
      "-14141543162859668839755449897447879139413790/439230920024096031841",
      "969311829106467648277685994914667431644985730661631415244063161039/9205329904679614020947695033361"
    ],
    [
      "40609666949051452717719717128373599536422254194/150704716996636936861225",
      "4512772611934484246203367073492670650599237833843128364159500221558827/58504634252118997510521691086599125"
    ]
  ],
  "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=-5088/9199 on branch u=38/17 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 13:42:44",
  "updated_at": "2026-09-02 13:42:44"
}