{
  "id": 6108,
  "curve_key": "1971590959448246241:-2867327899769454947374650705",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-41074811655171797",
    "3318666560927794288069165"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 126.44635612308461,
  "faltings_height": 8.458312946818314,
  "conductor": "195243106761946482026935784782118626756674",
  "bad_primes": [
    "2",
    "3",
    "47",
    "53",
    "4354411587536163121168111529999523323"
  ],
  "discriminant": "-322718899359454852096465799014587248142916564205568",
  "regulator": "7184308823105452.422823727165432454675399174899252455479928641767597563",
  "points": [
    [
      "-224576279",
      "1103154172676"
    ],
    [
      "-214651817",
      "1498533368984"
    ],
    [
      "-287741905/4",
      "19434125892415/8"
    ],
    [
      "62653951",
      "995517017936"
    ],
    [
      "72702745",
      "846544482500"
    ],
    [
      "88769791",
      "609854615216"
    ],
    [
      "97282651",
      "493382861012"
    ],
    [
      "121068035",
      "346884413644"
    ],
    [
      "203990095",
      "1851448500800"
    ],
    [
      "2493053743/4",
      "118610686900891/8"
    ],
    [
      "7760580811",
      "683427215880356"
    ],
    [
      "-228581276849/1681",
      "174221957875874992/68921"
    ],
    [
      "-41121726119/625",
      "37423454792495396/15625"
    ],
    [
      "14885922791/121",
      "475061067219856/1331"
    ],
    [
      "72461342479/625",
      "5296163912589392/15625"
    ],
    [
      "75424971943/169",
      "18883942603638464/2197"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,6,65,67,72,84,94,120,123,137: 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 02:29:47",
  "updated_at": "2026-10-29 02:29:47"
}