{
  "id": 507,
  "curve_key": "407492323275203415369568325193704845278076128999698942954641:-259543169754939136456911154299057571090771624357696161660478931820505438714115004488662585",
  "ainvs": [
    "1",
    "0",
    "0",
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    "300397187216364741269573095252831403479003263003057649505426359435866116343304926011841"
  ],
  "rank_lower_bound": 17,
  "torsion": [
    2
  ],
  "naive_height": 411.77211718372683,
  "faltings_height": 32.11793469252588,
  "conductor": "4234555647786645799327350113311740532609808811288652763141321341722263630766433744772980729646355774",
  "bad_primes": [
    "2",
    "3",
    "11",
    "17",
    "19",
    "31",
    "41",
    "43",
    "47",
    "53",
    "61",
    "79",
    "101",
    "109",
    "163",
    "191",
    "257",
    "557",
    "739",
    "3359",
    "24373",
    "29147",
    "32189",
    "36341",
    "281777",
    "51119741",
    "217991747",
    "714716037637249332961"
  ],
  "discriminant": "174444835767871731867585305526562595360481450229131311345771661442073342132614103498882080171116685157692129468670537641042631632916407111879665233542154442450528333497106432",
  "regulator": "19679648114471751314244490.0848763993086144339581803595679834742414174223",
  "points": [
    [
      "68100879821237655644088574078",
      "6171990469542346036871853558090242587114393"
    ],
    [
      "20117501624741157907686199148114398/323761",
      "659109289185302164419200624979115085923813719028097/184220009"
    ],
    [
      "2729658504974695036121678883166/81",
      "5286078240980427390182420491855361844161566769/729"
    ],
    [
      "117707198774883445219266459719364924/2002225",
      "6006251115394166518493315681440266867688338823562013/2833148375"
    ],
    [
      "-193199429173237784209989930020786600791634/2131643800225",
      "56133519433138693176243245214372140461728598887071764907814791/3112231922985503375"
    ],
    [
      "1883555812702124264108730301218",
      "2582006282651006485770604707596588979546110733"
    ],
    [
      "1856193076117758720351139086358330341445054/26569767140569",
      "951613159415471089611700163019000894761086696423557640919227259/136956176295804140003"
    ],
    [
      "390364594205967601930247611841433364399294/6878633971225",
      "21267308845328034838296064450407121170042532993178577172765723/18040696495841375875"
    ],
    [
      "93116988098286053302307781388030/1849",
      "61001565884921114376180961661929795272270944683/79507"
    ],
    [
      "16198034721898380982610500137152109395842007086/124298216485921",
      "2061547661577718547275083051086409879197557582769755471799700965158343/1385789753060265982031"
    ],
    [
      "3954500475807916340708978449562547436573699/49889444056516",
      "3947665034685631724974238032548868160782605587307141283969488061/352381416174410410936"
    ],
    [
      "227048889403184035990991755597386132766/2593763041",
      "1994733381695419054077343368260101797682390461878609439817/132097757915089"
    ],
    [
      "553848397453909874567380680065723649343927474/6998967657587889",
      "6523779269568144022384855804603809079936185508418218791736151382361/585532465406203976562537"
    ],
    [
      "-2336982434153565297720773626903547937894614/158544940480369",
      "41025574775412477779504440424404160840397696595460572014432822967/1996312751895768489847"
    ],
    [
      "-185400553674261013301529960715386739968877458/2825601053642689",
      "3601446653755294475591299353088989836706641108201201825529914910423/150198731812634321633887"
    ],
    [
      "-3483600326430932153384881834893062156218647339266/39005714789698491961",
      "4532861499219121601626652564325463461658565158851733596984675128085956395/243608457173930948440186486291"
    ],
    [
      "-826192788904714723695497162035470001762549970/7784089462335169",
      "1647857500129163940766536967949041806629990615557324570088008549783/686770651845013112590303"
    ]
  ],
  "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=13497/11680 on branch u=1/17 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 13:41:29",
  "updated_at": "2026-09-02 17:33:27"
}