{
  "id": 6518,
  "curve_key": "187346330037877881:-70243865923519471401177165",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-3903048542455789",
    "81300771720576338624569"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 119.31520599346848,
  "faltings_height": 7.931392027537,
  "conductor": "15320733234434998305899165043884892747847538306",
  "bad_primes": [
    "2",
    "31",
    "1472606987",
    "8625442757",
    "29953695619",
    "649485103870003"
  ],
  "discriminant": "949885460534969894965748232720863350366547374972",
  "regulator": "808186593358890.522626701110228905118722332701620105883170842334286145529",
  "points": [
    [
      "915046711",
      "27616349537789"
    ],
    [
      "2580414910965/16",
      "-4145098186868721497/64"
    ],
    [
      "52751829463/441",
      "10663686729111100/9261"
    ],
    [
      "78203843188/729",
      "-18644186326297301/19683"
    ],
    [
      "180395975841/10609",
      "153943683917350756/1092727"
    ],
    [
      "596470416273/43264",
      "1561473263589124465/8998912"
    ],
    [
      "-4595802830/11881",
      "372668307441648617/1295029"
    ],
    [
      "-70015369598/2209",
      "-43202746599592389/103823"
    ],
    [
      "-43711119905/729",
      "-6216142322205289/19683"
    ],
    [
      "3112946675/4",
      "173134454182421/8"
    ],
    [
      "6235975585017/32041",
      "14838167349291942737/5735339"
    ],
    [
      "24801841237035/96721",
      "120097628142410120708/30080231"
    ],
    [
      "21597682",
      "84122559645"
    ],
    [
      "68239201",
      "364275826019"
    ],
    [
      "-2005532039/64",
      "212865145081681/512"
    ],
    [
      "161975171",
      "1923113557884"
    ],
    [
      "48869937280/2809",
      "20337098362965039/148877"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,2,63,81,103,104,109,141,162,165: 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 >= 17: the 17 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 04:56:22",
  "updated_at": "2026-11-04 04:56:22"
}