{
  "id": 6528,
  "curve_key": "335352658746000:-6490530771479805585600",
  "ainvs": [
    "0",
    "0",
    "0",
    "-6986513723875",
    "7512188392916441650"
  ],
  "rank_lower_bound": 15,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 100.44926252643522,
  "faltings_height": 6.313105319078062,
  "conductor": "1596049724509937068078450914868353200",
  "bad_primes": [
    "2",
    "5",
    "3990124311274842670196127287170883"
  ],
  "discriminant": "-2553679559215899308925521463789365120000",
  "regulator": "199086301875.10336592241603605469819449792602143206459745284981178815",
  "points": [
    [
      "18349069",
      "-77828231528"
    ],
    [
      "44437789195",
      "-9367607073237530"
    ],
    [
      "-3566520455/1681",
      "-246431164883620/68921"
    ],
    [
      "9393549225/9604",
      "1195891918923715/941192"
    ],
    [
      "125932955/121",
      "1556874910670/1331"
    ],
    [
      "17478251005/11449",
      "778966954368040/1225043"
    ],
    [
      "2542349171/1369",
      "49169617878706/50653"
    ],
    [
      "26323816075/12769",
      "1973476087197190/1442897"
    ],
    [
      "898988049/121",
      "25453780510132/1331"
    ],
    [
      "-14643434741/5041",
      "649692784632502/357911"
    ],
    [
      "-1208583475/841",
      "-93140791776280/24389"
    ],
    [
      "14812513609/11664",
      "-1044756334599877/1259712"
    ],
    [
      "3961765",
      "-6481934120"
    ],
    [
      "134800968590/26569",
      "-43881466025649740/4330747"
    ],
    [
      "444296875897825/90839961",
      "-8229254171408749880180/865795668291"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,82,98,99,107,111,113,176,183: 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 >= 15: the 15 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 04:57:32",
  "updated_at": "2026-11-04 04:57:32"
}