{
  "id": 6020,
  "curve_key": "1351360374655760400:-1567618153954579318781424600",
  "ainvs": [
    "0",
    "0",
    "1",
    "-28153341138661675",
    "1814372863373355693034056"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 125.24293033169914,
  "faltings_height": 8.238329364768125,
  "conductor": "45610300416068014360512834699213194395225",
  "bad_primes": [
    "5",
    "7",
    "13",
    "67369",
    "28677907",
    "10377051661414102592492953"
  ],
  "discriminant": "6014854946596438188691553037496137569880717233125",
  "regulator": "329668009727457.4191354951278592022211293277874621828259299037639163391",
  "points": [
    [
      "-123400540",
      "1846457362067"
    ],
    [
      "88372111",
      "128670760878"
    ],
    [
      "91132779",
      "74532485345"
    ],
    [
      "92907804",
      "25979300420"
    ],
    [
      "92929210",
      "25034419182"
    ],
    [
      "93164510",
      "10664308557"
    ],
    [
      "100617345",
      "17127313157"
    ],
    [
      "102302560",
      "69953804567"
    ],
    [
      "103407925",
      "94105205642"
    ],
    [
      "126166505",
      "520270114682"
    ],
    [
      "135942635",
      "706687878942"
    ],
    [
      "148344115",
      "949972362657"
    ],
    [
      "4214684120",
      "273406218993407"
    ],
    [
      "2306331381/25",
      "5958020978529/125"
    ],
    [
      "12377548665/121",
      "92837699643342/1331"
    ],
    [
      "494570323225/5476",
      "37098606111773433/405224"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,8,19,21,28,33,44,55,65,77: 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": "2026-10-28 11:28:15",
  "updated_at": "2026-10-28 11:28:15"
}