{
  "id": 519,
  "curve_key": "11495905727768178677816878723254508157788561:24462869668046645023772037938700414482502707828291122175603013191",
  "ainvs": [
    "1",
    "0",
    "0",
    "-239498035995170389121184973401135586620595",
    "-28313506560239172481237594672791743924212306882379634187314463"
  ],
  "rank_lower_bound": 17,
  "torsion": [
    2
  ],
  "naive_height": 297.4516945621659,
  "faltings_height": 22.82605306166326,
  "conductor": "144709965057296981296158253023483904833485520354172293743617612639487430",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "17",
    "67",
    "73",
    "103",
    "167",
    "211",
    "227",
    "283",
    "401",
    "641",
    "1249",
    "2111",
    "2927",
    "21149",
    "141689",
    "206279",
    "1251697",
    "4221313",
    "29338571"
  ],
  "discriminant": "532881471847975695029613638600591195746203359824092309322330797911276879056397307228060361517620266723065553306624000000000000",
  "regulator": "617805698060403700269060.741775445861235363416455499189386143963329899255",
  "points": [
    [
      "67082540946176980152686/49",
      "16123588269568950299285884749369599/343"
    ],
    [
      "-330878192001158171236",
      "3834905613688862127829596701243"
    ],
    [
      "3991250292190875338566574/5041",
      "5971852659665526532196855483235650623/357911"
    ],
    [
      "-1640426409731912955556537707226/4485114841",
      "966600753450013937647570085375689291498078723/300372626016611"
    ],
    [
      "893211888284266062238293477625006/1526171156689",
      "10663651745561916434157924725593454291067390937231/1885405902063926887"
    ],
    [
      "-962606148419364293739577033906/2444214721",
      "268145583136572942577563968854265352964839367/120839531591519"
    ],
    [
      "-193497301751482070527587252214/1418049649",
      "72159800990354035431455585155661651889586649/53399495632393"
    ],
    [
      "-67822080430218074967498771137698/189668089081",
      "281385649413175851188629154227791904891587775045/82602159807577229"
    ],
    [
      "38021076259578053289210127736/52461049",
      "5080777738470239610609889823269694465772851/379975377907"
    ],
    [
      "-50752235791028341431066263074/183521209",
      "10180863869476930552618532289245278392531739/2486161818323"
    ],
    [
      "-23097704949766730580513346/175561",
      "70509206501662163917844587475352342787/73560059"
    ],
    [
      "68723916630070515882706501078327136/76475777071849",
      "14685196790390158717652826808293780032797201974103451/668783958951733594507"
    ],
    [
      "-156480279995244315883170873766/429276961",
      "28892695163989254330952356393162720415114847/8894189354959"
    ],
    [
      "-1729815191418444042571894/12769",
      "1850687764240362418271751164096822881/1442897"
    ],
    [
      "-71686243249330097920266282235721074/302981124760609",
      "20498168369937796428829012357350550023392825275778761/5273796409920200748977"
    ],
    [
      "23479983006374107156150430703029006/43313865080929",
      "306488699331159523067507542615824841949550101986519/285062709731475212783"
    ],
    [
      "-509678515252760949437014560900706/2257960027801",
      "12806269124173282759768899848056741517609919706123/3392925893735200451"
    ]
  ],
  "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=-2346/7631 on branch u=38/17 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 13:42:40",
  "updated_at": "2026-09-02 13:42:40"
}