{
  "id": 515,
  "curve_key": "74282057578907178962574292243150496669415038746801:-624986730969629077219046118607132167387458696424003886470278977671542083945",
  "ainvs": [
    "1",
    "0",
    "0",
    "-1547542866227232895053631088398968680612813307225",
    "723364271955589209744266212018571526651261866391858242759877284200212921"
  ],
  "rank_lower_bound": 17,
  "torsion": [
    2
  ],
  "naive_height": 344.4958617003703,
  "faltings_height": 26.61210216017743,
  "conductor": "71504239127973300880966109863118283531248692050825111681599812381652833211716013826",
  "bad_primes": [
    "2",
    "3",
    "7",
    "11",
    "13",
    "17",
    "19",
    "29",
    "41",
    "43",
    "71",
    "89",
    "101",
    "157",
    "233",
    "419",
    "431",
    "1373",
    "1489",
    "2243",
    "5867",
    "7103",
    "37529",
    "55009",
    "3786919",
    "1286020398220673174161"
  ],
  "discriminant": "11149833330883655057294100720239245622463300761506897035376857753438192206974460580710806177679164191158940834121380769261497161437078744422715392",
  "regulator": "588801609565891721391078.732075450745452885892768142880957946014021560455",
  "points": [
    [
      "2134086120281460864504615402405531489/2643902016064",
      "48148218736640233823336654612253225234052258611483117/4299005829336192512"
    ],
    [
      "9625830693395214085237779997218/9006001",
      "14562559957680533577238316386277541746952158619/27027009001"
    ],
    [
      "5853403402092197864414341619046627498/6966363735769",
      "2345171968661810616726327728440819636664917571003317497/18386929881460133603"
    ],
    [
      "-55170110527911587732533978467962586/66357244801",
      "20478774056569302610917392249098549399922100426835365/17093559903492799"
    ],
    [
      "70971199336520428510635551573298/86136961",
      "69844674559146735257619892810644176941322913483/799437135041"
    ],
    [
      "8905851505959445958120243224915841274/10911625725625",
      "2276465858606899266780434884155400883340525642195643593/36044100468813921875"
    ],
    [
      "750081415113176027169535299761611401150/1198678937095249",
      "112890644931050932931548195265209361231161763968334113473/41500593835412296448807"
    ],
    [
      "20648704291817694941418809933191567329762/25294267746308089",
      "256351910896141124409275391969396286077923308068379515922407/4022843772630558885548813"
    ],
    [
      "2707015916409867894085411861594594602/2566554590209",
      "2114491033255623464409277013564792845877670449330315229/4111741081580557823"
    ],
    [
      "608679630024355243332626286415429386/678160485025",
      "133829408476909832513512755911956988029087484235362451/558468550220512625"
    ],
    [
      "1366393654474900058885295283953299491506/14938090906719289",
      "1393537415020805539716394352961381404249449593848912039214297/1825755623808046166285587"
    ],
    [
      "-740611176434284625050048044526204303321344/517051938804918169",
      "13063715974301489675764177439021760054564060851723820612274815/371793040814190370121253053"
    ],
    [
      "68307895556107052650676356084272966189606/81125775651605161",
      "3037528595698784250878682388679124136279317428635001510286039/23106719548267481226709909"
    ],
    [
      "-39407730953702469614737723880991721508925414/28649018760067174225",
      "76582354173672213755261651057419217225048465528537018313386422821/153343237334778357949052318375"
    ],
    [
      "38812780389114554934455942192795220/9409582009",
      "7331667319075941140264470281801597331486076210854617/912757683619027"
    ],
    [
      "327040435295581792078536018251759300427234/398681135212008289",
      "19321084783904606577224295717201077209577806263726443967554643/251732059724063876378208913"
    ],
    [
      "-349911680318269039975771864667263026/1350636784561",
      "1651437414090946138892785851048604438931210489355826275/1569668201276472809"
    ]
  ],
  "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=-8631/24595 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 13:42:24",
  "updated_at": "2026-09-02 13:42:24"
}