{
  "id": 525,
  "curve_key": "26493470912447010810795998142512598294100279313401:-135733975920780840404383329897706623310572346963485157969457871748128830605",
  "ainvs": [
    "1",
    "0",
    "1",
    "-551947310675979391891583294635679131127089152363",
    "157099509167570417134702882126736405833398299501722157639537535414016042"
  ],
  "rank_lower_bound": 17,
  "torsion": [
    2
  ],
  "naive_height": 341.40294835706186,
  "faltings_height": 26.28302072041932,
  "conductor": "15677359235871344955055989853595029670413032016398269566737332835394059780410056266474",
  "bad_primes": [
    "2",
    "3",
    "11",
    "13",
    "17",
    "23",
    "31",
    "61",
    "73",
    "97",
    "181",
    "233",
    "353",
    "419",
    "449",
    "457",
    "557",
    "941",
    "1531",
    "2087",
    "2251",
    "3307",
    "162517",
    "169019",
    "277919",
    "1650023",
    "6171167",
    "2813681711"
  ],
  "discriminant": "99630216585965563598177951756063565243354960961544175018243802779459203248698726276377984776784433760383995041597009832908919060277911778018492",
  "regulator": "1030657730556041331249751.87568581700702543101947176445781723721892582014",
  "points": [
    [
      "530439955155441135430519",
      "116501761790458205944436753284846142"
    ],
    [
      "147642799367561458262364527249/219961",
      "30781715013266249687602446457234397845570062/103161709"
    ],
    [
      "4292713922607666691092133/9",
      "1308267401128264967879610658050145832/27"
    ],
    [
      "5463996321035297078449925108651489/4242828769",
      "347617615793941222492603072890983839553137641116670/276365137526353"
    ],
    [
      "1635396179182666283380456747755523939457/9637285482284329",
      "247296132293975656074224279159103076292041693245410926215678/946089202952641800091733"
    ],
    [
      "-3550972949807821414401798268044599851825/15631955523896761",
      "1016976695486939709544597123019341615919856058442986234412454/1954429305857814408994259"
    ],
    [
      "3757902777998162381532693903827313602/457797032449",
      "7255952719857618437627362965015786757071331152137431548/309748676734220543"
    ],
    [
      "1568730732295574366536366569118684874351390/5285081108186402329",
      "1693187841847535081963155698694857744783144332174086036111081704/12150033768772599101801764483"
    ],
    [
      "920391244685888426773268074488265559/2512481776561",
      "253933189386469422479171843721525436020611482371772422/3982487126873086441"
    ],
    [
      "-27354945537009754840041229217503/37356544",
      "93757279514796413906046308627533424873537221215/228323196928"
    ],
    [
      "7869949428401723243837327921203270232593133/15275163351416550409",
      "5815615199574752954640868577318134959093495812604530392142982888/59700613609173800649806096923"
    ],
    [
      "2234690241514954804154317910/6241",
      "36143584355482685895650701314076213689207/493039"
    ],
    [
      "3146428045934368502262259464994380997967/7808447628507241",
      "7517233256909655082309908751988602154012836533924668695446/689996762049493952513461"
    ],
    [
      "309720011469948671824265909543104592/768040151641",
      "6690866886412050397741238293190091931774605444202059/673094260054987939"
    ],
    [
      "30698919756697316645913131328346235/26057062084",
      "4495129274571179038637104367257999274542556318160129/4206183075723448"
    ],
    [
      "784766019513518644679924787969857837945/1271500594823449",
      "10294049364400117661207710645832281312363780251948003207738/45339304260778190551043"
    ],
    [
      "2331099338891090847585562039759301/377175241",
      "111770479682686037400004338227220520295680572208728/7325120355461"
    ]
  ],
  "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=773/4404 on branch u=62/29 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 13:43:11",
  "updated_at": "2026-09-02 13:43:11"
}