{
  "id": 6521,
  "curve_key": "35640255143078883921:-219113272299238557035644650969",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-742505315480810082",
    "253603324606041325439235681"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 135.11877266410323,
  "faltings_height": 9.17276875228672,
  "conductor": "8620128113948600358915924702619189084210",
  "bad_primes": [
    "2",
    "5",
    "7",
    "23",
    "97",
    "1249",
    "1921435612304048017245320365219"
  ],
  "discriminant": "-1585291487937183596297917393847546442929394744320000000",
  "regulator": "596789843420.3530397401681335789384738150251392745911948683668398942616",
  "points": [
    [
      "-92944719375/121",
      "25627727311343317/1331"
    ],
    [
      "3340395841",
      "-187208226573921"
    ],
    [
      "-841149919",
      "16823622641119"
    ],
    [
      "29927084209/49",
      "-1813145318102903/343"
    ],
    [
      "65378488761/121",
      "-4242200294112851/1331"
    ],
    [
      "3616303969/9",
      "-121143686694667/27"
    ],
    [
      "254399413381/441",
      "38454855498585119/9261"
    ],
    [
      "26694609751/36",
      "2272997365395989/216"
    ],
    [
      "23886080997841/28561",
      "71195777591322505191/4826809"
    ],
    [
      "182231649049/441",
      "38564408873425331/9261"
    ],
    [
      "14532972289/64",
      "-5035026367786317/512"
    ],
    [
      "643641001779/1444",
      "183634802301405763/54872"
    ],
    [
      "953863307806089/2181529",
      "11409612575787502138707/3222118333"
    ],
    [
      "-986570159",
      "5088496268079"
    ],
    [
      "1411810401361/3721",
      "1168502177127518499/226981"
    ],
    [
      "403301121",
      "4443642331359"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,11,12,42,55,72,126,144,169,197: 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 04:56:43",
  "updated_at": "2026-11-04 04:56:43"
}