{
  "id": 513,
  "curve_key": "41020217263695368494857128360242614008525103825:-7424486657276578032227915965867236669394925723732640455941686264425625",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-854587859660320176976190174171721125177606330",
    "8593155853329372722486227496348290854806889561122987902436023874297"
  ],
  "rank_lower_bound": 17,
  "torsion": [
    2
  ],
  "naive_height": 321.99118270167935,
  "faltings_height": 24.809516165644837,
  "conductor": "28977213133631261182797956136132989566312790364730373629599921717272285151559950",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "29",
    "37",
    "53",
    "79",
    "83",
    "107",
    "151",
    "179",
    "193",
    "223",
    "227",
    "349",
    "467",
    "479",
    "521",
    "647",
    "761",
    "863",
    "1321",
    "1367",
    "1567",
    "2957",
    "334993",
    "9836003",
    "1166024137"
  ],
  "discriminant": "8043983691981932930289123818275852398131566994856457857097032952126234370293978013135704388682314250301287716342561728893350792256000000",
  "regulator": "5994334272502409943.46890884001842830885301136751995834060434981185804074",
  "points": [
    [
      "133665179871590627347839",
      "-47775280780823544998811447350749895"
    ],
    [
      "11230897903166785077513",
      "641836784448154607364054604299415"
    ],
    [
      "2562893425126927154199249",
      "-4102680312768652062532240810554966425"
    ],
    [
      "-20830643533980545338311",
      "-4166056582268379948190523433484145"
    ],
    [
      "1120680038481368350580685",
      "-1185976200675693487177417325258540129"
    ],
    [
      "-19109467362069526476501",
      "-4236229120772000742708215701299425"
    ],
    [
      "-24692726000744577874353381/1156",
      "162537173098232871511026716661321703225/39304"
    ],
    [
      "194496920415822063016839",
      "-84852887842176004925542289614583945"
    ],
    [
      "585454701088453215923917/9",
      "407930273364925394207010525077364917/27"
    ],
    [
      "-29633966617293892201311",
      "-2809673449572184350313954850217545"
    ],
    [
      "1624818851228312885074687/49",
      "1402372017323576974829233915240845103/343"
    ],
    [
      "328899661320623196004569",
      "187899537405379437789477382195700215"
    ],
    [
      "21891309646065193697439",
      "613244173985668446149171568238855"
    ],
    [
      "55407365073189139347427689/961",
      "-366055458437090236523892378625928341175/29791"
    ],
    [
      "6451144352471994784114791/2209",
      "256892847460951506385528611098287247225/103823"
    ],
    [
      "206819176004087011564400239/17161",
      "-473442154494027045249311832235248579925/2248091"
    ],
    [
      "10214415042700802605905",
      "964237703524679874390966614591431"
    ]
  ],
  "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=-3688/7737 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 13:42:13",
  "updated_at": "2026-09-02 13:42:13"
}