{
  "id": 6524,
  "curve_key": "3729710685390616641:-7194272910342338347079817825",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-77702305945637847",
    "8326704776766245869603695"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 128.28858702090554,
  "faltings_height": 8.471153187798029,
  "conductor": "30511387425960094231483481326456544379926",
  "bad_primes": [
    "2",
    "7",
    "11",
    "18011444761487658932398749307235268229"
  ],
  "discriminant": "72615516458003271976753663823409002330525544005632",
  "regulator": "6139335829465.943556578719643844986571632300830319212714488896370664550",
  "points": [
    [
      "97709271103/324",
      "-20471530511280061/5832"
    ],
    [
      "141867003",
      "398127219774"
    ],
    [
      "165853251",
      "-41313739458"
    ],
    [
      "676040399/4",
      "1183343020591/8"
    ],
    [
      "183599095571/961",
      "20092840788871042/29791"
    ],
    [
      "259833607003/1849",
      "-33975211828320486/79507"
    ],
    [
      "3680731739/25",
      "-34933240779322/125"
    ],
    [
      "1240888691707/361",
      "1377867920768887338/6859"
    ],
    [
      "95367118941/625",
      "-2370397929275064/15625"
    ],
    [
      "3161297131/25",
      "90397205904806/125"
    ],
    [
      "-321056381",
      "424345766590"
    ],
    [
      "38809440547/289",
      "-2752714303005538/4913"
    ],
    [
      "-3321817",
      "2929981347374"
    ],
    [
      "-25609953209/81",
      "-827192245401602/729"
    ],
    [
      "123239625",
      "788903386170"
    ],
    [
      "-5693995802791/27889",
      "18443353077330768758/4657463"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,18,21,26,31,43,55,133,134,177: 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); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 16: the 16 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.",
  "created_at": "2026-11-04 04:57:04",
  "updated_at": "2026-11-04 04:57:04"
}