{
  "id": 518,
  "curve_key": "33668616055418137021746312547762715709243896473776721:-6097189566758591058137940440854684688625815065958960118922930767587270966199769",
  "ainvs": [
    "1",
    "0",
    "0",
    "-701429501154544521286381511411723243942581176537015",
    "7056932368933554465437431007351574774882649922846633445924603245047947055017"
  ],
  "rank_lower_bound": 17,
  "torsion": [
    2
  ],
  "naive_height": 362.8452176154881,
  "faltings_height": 28.11399584895546,
  "conductor": "132399771304839820510988825724471544529303829934050389830244032069096333604610",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "13",
    "17",
    "19",
    "29",
    "47",
    "61",
    "67",
    "163",
    "173",
    "193",
    "229",
    "367",
    "397",
    "461",
    "593",
    "761",
    "1021",
    "1399",
    "1697",
    "3671",
    "4021",
    "6101",
    "6761",
    "7487",
    "7723",
    "43517",
    "122263"
  ],
  "discriminant": "573035091826912912426444929148610619989680402756691849065020235490250642686808109945781778071288280003979030330566301364914248431896121963199910272000000",
  "regulator": "362658138559063799071887.178764969600814616968748220327214177197924360989",
  "points": [
    [
      "-635911889648194668547866",
      "86618257751572382574991932645786322773"
    ],
    [
      "222388385639007938555705104596/16129",
      "5353469910789300734649643097483406214105929/2048383"
    ],
    [
      "-19520952108308752460002947683376/3500641",
      "680506348142281681908942673585674920265655711683/6549699311"
    ],
    [
      "-11344135338019654957601403066/1681",
      "7385539168276530272460254785610883689804013/68921"
    ],
    [
      "86695319269884033308584154807697414/3664970521",
      "13498060594434793208815649000903916566275309898831487/221873650370819"
    ],
    [
      "54766057794814753361738608139616/3988009",
      "30035495100610414822141724947990410101538504849/7964053973"
    ],
    [
      "807185079441655471108903641667368294/33504007681",
      "394678962485402835459150523033220985896737807899269973/6132607069937921"
    ],
    [
      "143547174632379830198608674",
      "1692415976615822619090017779849102949613"
    ],
    [
      "-1418820961063097266925636424138/54289",
      "1098246284057754141915665922242050935991964877/12649337"
    ],
    [
      "3959820474045962728781736428576702262654/2206895935267321",
      "7898446042621360788214982452528123879869171571738116634736703/103674698764469386950131"
    ],
    [
      "54883795450668820190156626093195582672811498934/3148693315230784875841",
      "1986989922622262582242897138382522921103889721927700161052079736105493/176683316682895632330564861819361"
    ],
    [
      "629364033179767173664052750209514329466754054/35872557441012677641",
      "2640635783598702756864666153138417925653271554565860021685159751703/214854032669196911698379574811"
    ],
    [
      "60928563784495983740712089068915851123306/4432162319313169",
      "1034866818339089237399621675102852491408218079415253944475339/295068932709004418519303"
    ],
    [
      "-361156938779132463605738853748798169614961286/259862778566497094641",
      "375361200605550206988644148685670305158050434705257744590556878120667/4189055523370964248311378208439"
    ],
    [
      "66496302586194963312348153149967774311099470986/281249773656912504889",
      "17044004408884177386993671823838709190346370709028977454201016412233031/4716700196206303347502227072787"
    ],
    [
      "634512268928943752065103107362265965893046/4306882522146649",
      "497761467150198402915778816521721448749905567569683011927153951/282647103100362156899107"
    ],
    [
      "334977826557620944448175245391800355816/11229918125449",
      "4236812744286246283952227295439570625735324704754961915289/37632657239619022043"
    ]
  ],
  "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=-8433/8669 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 13:42:36",
  "updated_at": "2026-09-02 13:42:36"
}