{
  "id": 509,
  "curve_key": "2603700890341668955046756306230366149261954488656:-4133260504920736310403762155943375449819684782584732366917686384367285696",
  "ainvs": [
    "0",
    "-1",
    "0",
    "-54243768548784769896807423046465961442957385180",
    "4783866325139741100004372428412867476659045211947418691104840226960900"
  ],
  "rank_lower_bound": 17,
  "torsion": [
    2
  ],
  "naive_height": 334.44305494800835,
  "faltings_height": 25.75679935633105,
  "conductor": "3018786376446535998275609684836898200968187274821185538667187516489187463931509795160",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "19",
    "31",
    "37",
    "61",
    "71",
    "137",
    "151",
    "163",
    "179",
    "383",
    "499",
    "1087",
    "2083",
    "2269",
    "3331",
    "7211",
    "8101",
    "13633",
    "124853",
    "154571",
    "19575161",
    "28502407",
    "204479773"
  ],
  "discriminant": "328309339512561184929823039121289205321653805939380365592608577363494992384734816174565567831914060926701719794507135024647495552726644000000",
  "regulator": "10528819855347700003385602.7112245079341167105026905493783189957591991576",
  "points": [
    [
      "12948908284061969468667700558785488636/64028947272481",
      "23395272679586428591994040289181753618866578303745085850/512347406545463918129"
    ],
    [
      "126920620817053040563788875968659733/1058433555204",
      "2057600290631114775061162698004273515250728690534979/1088918558460985608"
    ],
    [
      "1340895516564754455403705980200560/9032031369",
      "1474360572912478632831540325426494817849696491130/858377165215653"
    ],
    [
      "51420441978863500784181027826367100/320680966369",
      "2623836834040354042787624538796141567243689459334490/181597462402201903"
    ],
    [
      "2017980057769107278571659939112160/2072707729",
      "88261199731994890720906370465948276119300382530770/94364164778183"
    ],
    [
      "15272560156828376549023851773063330/406856449609",
      "13733597130550255324257402891799909688591170315844460/259514606952449477"
    ],
    [
      "-42380237404833838504520084859037891960/9093350896237969",
      "61539428302674760440566802301156703659649113740446452882650/867133420995462948863047"
    ],
    [
      "49170067264690379086597244981572720/422108790601",
      "1856249851348837061163333540201197547475511595753030/274243659144679099"
    ],
    [
      "492452493217031601797500104830215407947824/7445728525419495529",
      "783078784595781009416404028571310024636099887314193956665743270/20317056877293196847270379083"
    ],
    [
      "1705776007848404430792355724299657740/6615538940761",
      "1516185804910446149389134242008733836166603710525574690/17015622627824204509"
    ],
    [
      "21773858050553270907419633334773600/103207345081",
      "1732440730878809532957745056117865817510766346186170/33156288473376979"
    ],
    [
      "1475685258394958128330998508387132210069984848273940/243377377056059541872802289",
      "56646704590172159122039064521417205713067346026584713834774994289734049998350/3796822630370410678999430127804614044087"
    ],
    [
      "5924958301318129902827866499976941425395090/39924989526634308721",
      "379715087177825356831906633793675802620229814761494360936438860/252270934699831531231763990519"
    ],
    [
      "3841969347544663602368115523008310/34188379801",
      "65477120919287689341314156166747773161669459831500/6321465613584701"
    ],
    [
      "245093070214777709265279491857708/4779924769",
      "15277930657015370780252594305939904817352082280534/330469658754353"
    ],
    [
      "2534636704535988210570337356239931356877607656/22151070891725150005801",
      "28588864245004191827823581470513137302184675185684799239362553683090/3296796120771340504479153860782549"
    ],
    [
      "13398909391752369672630579653694402428786637760/156758532098236501571161",
      "1724317107749622310277718132753388046700905690627669049246928483724550/62065003193967514189101259853890909"
    ]
  ],
  "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=11/19779 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 13:41:37",
  "updated_at": "2026-09-02 17:33:24"
}