{
  "id": 511,
  "curve_key": "646847434197990730794123859350583467218944049121:-514846526115923802788225829591930286538059069974145246670437567168416081",
  "ainvs": [
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  ],
  "rank_lower_bound": 17,
  "torsion": [
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  "naive_height": 330.26531893943275,
  "faltings_height": 25.38884942771552,
  "conductor": "371813998023695463104408943380209742606395805686060389724134210913039287870",
  "bad_primes": [
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    "11",
    "13",
    "17",
    "19",
    "47",
    "53",
    "79",
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  "discriminant": "3230050039767995941465304682565004975933777239625282300706909439707303788909135877913986698890718416408006853165474706617395712092723200000",
  "regulator": "47117307022781189875744592.8533325600483036021988483838738388934447825977",
  "points": [
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      "140488965594723648072863762497557404125817724649126439548913949356820208/52752638127975512652491069100765354129"
    ],
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      "197238624811856892144290728557055789053/4485936210945424",
      "2824462640821124709457044448801952305624515133750884563293/300455141744071508043968"
    ],
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      "4765567644683118326860826313712103006750573/18641279563259464969",
      "9474838794275858070753388996340227815403768154481081179794541172/80484746298829440944710845797"
    ],
    [
      "-165299580979518469020951719304724473755673703713961/1933715233774852261197674401",
      "2849822260295145765060789243731037679636155982335684317668241225608137051528/85033249927866820445946101217037871728399"
    ],
    [
      "22044492163037832050645044692406903358839053687/412478145459248037330841",
      "1409927766063264294748388912692297202584120197138960952780174863473536/264911867714426887706733412588620611"
    ]
  ],
  "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=3611/15087 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 13:41:46",
  "updated_at": "2026-09-02 13:41:46"
}