{
  "id": 517,
  "curve_key": "224460008292768247600609149992436511976044751139423121:-106296650048249159974749792108753814529049782246200157377888682351254289385313465",
  "ainvs": [
    "1",
    "0",
    "0",
    "-4676250172766005158346023958175760666167598982071315",
    "123028530148436527748552999736184956122566447737600037487539252762948104949281"
  ],
  "rank_lower_bound": 17,
  "torsion": [
    2
  ],
  "naive_height": 368.5366118916896,
  "faltings_height": 28.455273065025807,
  "conductor": "16502786707526974416379816040011713123070308567840902601861766573584920924211702",
  "bad_primes": [
    "2",
    "3",
    "7",
    "11",
    "13",
    "19",
    "23",
    "43",
    "79",
    "293",
    "397",
    "641",
    "709",
    "761",
    "997",
    "37589",
    "51169",
    "121843",
    "297457",
    "764459",
    "5968009",
    "9274813",
    "18096011"
  ],
  "discriminant": "5690170417883235106130043656047492632420738141033822025180127320113421585670542023422479853407921889782144369545376498760077066396959303669428059954163712",
  "regulator": "3587994529051353905493328.09935679242622696341833954454287177077473762185",
  "points": [
    [
      "-21759388581446599346177479083428787/2638260496",
      "54379759108214908998006549389965754284286269047579777/135511612116544"
    ],
    [
      "242884560082604191251594677731468087873144826/5970177811478562361",
      "159353183614397221825289498143871046392317886313943710563457548963/14587501007699830592322319891"
    ],
    [
      "61302708085872462553303757920643916526974/908515990276801",
      "9274679296339698780497655289510075401694103742376899162716679/27384124664011234744799"
    ],
    [
      "3159075331767571239696072838861359438/48999806881",
      "3245307320840728988818042919008137900629104054830792471/10846548251371279"
    ],
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      "16585615723118317487276365502715922686/449464317241",
      "8073204696584738357499944839621459160000392605144169221/301330317029028461"
    ],
    [
      "44382449276073154064796876",
      "53941845463136057357924318122521461481"
    ],
    [
      "993208226272286906633106785496989080175574/43733466803481361",
      "1545120863981906323235186910900959520103362789787282224890166681/9145782281075572093659641"
    ],
    [
      "24622352339622117148012956636503137470174/610176897694081",
      "91885411565849227222374588601188275750598389282549594793399/15072442674206844588479"
    ],
    [
      "354697055847627337115648187091569774/9339482881",
      "13097350331207266964267309943676973331521528409332601/902576965102721"
    ],
    [
      "43151829798668508577761758233532261846910/1061125608455929",
      "369854648503852656237560992819440205398588444477372008347707/34566084988780036068467"
    ],
    [
      "3249322786421981947967994313281531242478/80925155297281",
      "273170001962305658356458392200077364174972246944826992153/727989829954647608321"
    ],
    [
      "87489901326724262240504762366391778588857626370/92127961767303452809",
      "25813052479496367098808780134944019235675208393670188737633003126049473/884274689547131985662441982373"
    ],
    [
      "12184682662779026408862451273708729470/684949898689",
      "120880847284064066854551090600379738440763064280636901497/566876180303294113"
    ],
    [
      "562646020201283875413371374565915435405235/6859014256935556",
      "306652098900573601384209588975583175838081575061893628251496681/568057840341512463666296"
    ],
    [
      "44051081463650394831049730520052415305208460174/1152124410212886661489",
      "443360090740945209498148350227817882342600565807278706214839301659167/39106510677469550078639603514487"
    ],
    [
      "123972536396726656873006947463764/3478225",
      "262306681831455999499157194310814073030353310683/6486889625"
    ],
    [
      "8610051128604955445968251584382445868175067902943903458/223679570653952287367897596921",
      "827480349650422197473435477466147538959008294783149324539179472108168438226506231/105788748816096133649858681981590781077055219"
    ]
  ],
  "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 3). Fiber t=-20341/26177 on branch u=31/17 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 13:42:32",
  "updated_at": "2026-09-02 13:42:32"
}