{
  "id": 514,
  "curve_key": "465055161454265873834038264872433623932889075201:-320477055236579985659369303624294922690280177979675460952741900603528449",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-9688649196963872371542463851509033831935189067",
    "370922517634930538957605782838566512600361610129476142250057405792091"
  ],
  "rank_lower_bound": 17,
  "torsion": [
    2
  ],
  "naive_height": 329.2963641997233,
  "faltings_height": 25.3080867976733,
  "conductor": "78193173609204889224508498376195855635709965623422249377518655313399916501570",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "31",
    "37",
    "47",
    "83",
    "131",
    "151",
    "199",
    "271",
    "313",
    "523",
    "619",
    "787",
    "1097",
    "1427",
    "2423",
    "3023",
    "4229",
    "5639",
    "5821",
    "15959",
    "27809",
    "140593",
    "175261"
  ],
  "discriminant": "-1229821661984755649934627616974482983435721592962363255355131867011699297303270922214609159302842459728454154044639338625455638114304000000",
  "regulator": "16795087060466347431630591.3477608992943958968043482271203396013210727801",
  "points": [
    [
      "4948100799489570488195534171/75076",
      "88855673559939611474089651586316074794161/20570824"
    ],
    [
      "1709633897326706447636794692491/30129121",
      "324866244380159356910787339599136172054669866/165378745169"
    ],
    [
      "179373141192472767882689018926163771/85362724561",
      "75887076588748343559124754327891340090954997396881306/24940341872262809"
    ],
    [
      "48692652566990898686460188721719572162931/832031087712779161",
      "1583145622120990001120548880078313141608910271627697408383074/758943297878669919729137341"
    ],
    [
      "15130936858205526007443702690791491/162616208049",
      "1087454137224438783992262445638333921941333781108818/65576124209215593"
    ],
    [
      "-644467921754817082454477574277977897301/8985711521898049",
      "22485669889459769876851616925952008000047652921846269843658/851782489395301105570657"
    ],
    [
      "213776206021076268303446513497409771/3063448573441",
      "31557118729179420447653335440275355293011988823391734/5361865198085152511"
    ],
    [
      "-5378495996768520579524963830672972173541/65455652417778529",
      "414361439867706361442402421101627957658914755333931585201238/16746371987001717803853167"
    ],
    [
      "309599263560942613309718875557331147811481971/4829845427982889243801",
      "1222199398582022543670185181863246086355844431552380353759859141986/335660199250385998619609141375549"
    ],
    [
      "748440433178611758685376771030677131/13331151904225",
      "96587311660057368248296890344671566514613840218604074/48674501865427756625"
    ],
    [
      "785915685308898904035588756333827339691/4481835335073121",
      "19127820762260694692005311228413423510723827930993369337334/300043238559661380830831"
    ],
    [
      "290168786589308433019102050606599179109771/3163273116031236961",
      "89672469894206138354195610885476850543363317476890111365732394/5626068758855146382896785041"
    ],
    [
      "5992197495574655249119020843878467488241831/180682599741282081",
      "14668215015489270419076582596505181297109023226788655847733687426/76802347762320753957526479"
    ],
    [
      "67835069115243942163050056035/514089",
      "13742202879883883887773222902607825622934866/368601813"
    ],
    [
      "2723694627292897774204961217726808459499/78953293482046369",
      "195607456190039898634445379849297679545764573881980657103542/22184792781941510387141903"
    ],
    [
      "90028927537565880432830307144243665867815331/2144151943870996282276",
      "613155498506454125098303607592393665125256787478766914095299863899/99284930730116690100779580709976"
    ],
    [
      "2033794526461466710263145240841495493886136411/10218529531833223441",
      "91719263386849835553276708585594035064103111294185772032861345848646/32664995792903596628222591239"
    ]
  ],
  "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=-5137/14798 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 13:42:20",
  "updated_at": "2026-09-02 13:42:20"
}