{
  "id": 6180,
  "curve_key": "3869588738271524961:-7644941155121690647546259985",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-80616432047323437",
    "8848311542285694409453845"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 128.4076833060377,
  "faltings_height": 8.530356672084489,
  "conductor": "1306046612716079154116875733335056508018",
  "bad_primes": [
    "2",
    "11",
    "13",
    "23",
    "47",
    "11689",
    "47544887206007",
    "691023926341891"
  ],
  "discriminant": "-291087140633757129466097043550694728265683623477248",
  "regulator": "1171929037758307.608926218121939282658429744877497697969782811540851053",
  "points": [
    [
      "-327277805",
      "421240147480"
    ],
    [
      "-315866097",
      "1672853144664"
    ],
    [
      "-141003123",
      "4172848106700"
    ],
    [
      "8058881",
      "2863412932562"
    ],
    [
      "18917961",
      "2707384778922"
    ],
    [
      "153078483",
      "307757596952"
    ],
    [
      "450039901",
      "7982064269452"
    ],
    [
      "494021007",
      "9465019201560"
    ],
    [
      "518359373",
      "10311934869964"
    ],
    [
      "1556085567",
      "60425304087080"
    ],
    [
      "2888971539",
      "154555206841880"
    ],
    [
      "-234917999369/729",
      "22987641866625800/19683"
    ],
    [
      "3797983671/25",
      "40943328065016/125"
    ],
    [
      "4329115015/9",
      "243539326152200/27"
    ],
    [
      "29156670559/625",
      "35592500406908952/15625"
    ],
    [
      "312612939103/1849",
      "18002557127811984/79507"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,5,60,63,72,75,77,101,113,120: 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": "2026-10-30 14:40:47",
  "updated_at": "2026-10-30 14:40:47"
}