{
  "id": 6133,
  "curve_key": "4029313967810981361:-8207995078608632434285540425",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-83944040996062112",
    "9499994324931186677586755"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 128.5498123993655,
  "faltings_height": 8.594151493596758,
  "conductor": "66990644961370594294331709277299659581362",
  "bad_primes": [
    "2",
    "3",
    "7",
    "113",
    "28031",
    "39359",
    "4264641446277104227915783831"
  ],
  "discriminant": "-1130657327131201185816591008387567774008199833179648",
  "regulator": "66192228708206164744.01302663565167163332924261104194115541626854222656",
  "points": [
    [
      "32753253111",
      "5927390347072807"
    ],
    [
      "-937402001232768947/3887148409",
      "960888584814765588277706875/242352041855923"
    ],
    [
      "-15056216822864753/44930209",
      "760568644632612905919/301167190927"
    ],
    [
      "-2607355206899/15625",
      "8482470823398239349/1953125"
    ],
    [
      "-842074938723/3481",
      "812520832013488475/205379"
    ],
    [
      "-5031536411/49",
      "1415782073681151/343"
    ],
    [
      "142644918345/529",
      "30949101087330815/12167"
    ],
    [
      "215440281789/1681",
      "63413527868740745/68921"
    ],
    [
      "8688404949349/55696",
      "5894484992382170075/13144256"
    ],
    [
      "118353449053797/11449",
      "1287064961429413248611/1225043"
    ],
    [
      "251594255169283/1515361",
      "696676298090918802429/1865409391"
    ],
    [
      "626228432117797/12737761",
      "106535702907226094394105/45461069009"
    ],
    [
      "1540707762559143/23843689",
      "242704239418814975499989/116428733387"
    ],
    [
      "7820426705091153/17447329",
      "573488401815656517972421/72877493233"
    ],
    [
      "2393739263501443317/29739347401",
      "9266447711246494693432832075/5128580198649851"
    ],
    [
      "64533457481690024085/493974225889",
      "303261427280035062248193941825/347181387104243537"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,8,13,24,66,72,95,107,116,139: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-29 12:56:18",
  "updated_at": "2026-10-29 12:56:18"
}