{
  "id": 6507,
  "curve_key": "67221844439727472102881:-17449431474893961709397667608907345",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-1400455092494322335477",
    "20196101244440347233074584760845"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 157.68923027337178,
  "faltings_height": 10.920464201982552,
  "conductor": "50386897596845092487449703156078065833519331942554",
  "bad_primes": [
    "2",
    "3",
    "11",
    "13",
    "41",
    "476122015347169",
    "77137152711882516579848849023"
  ],
  "discriminant": "-417926274943831303860498758350837905008872589937060969370288128",
  "regulator": "9359673150175581.5251386652917547962895854892970966980134140822639301694753",
  "points": [
    [
      "151089110623",
      "-57075925241481328"
    ],
    [
      "33749720650591",
      "-196067243990509759984"
    ],
    [
      "-1614436001177/121",
      "-8041959225514855376/1331"
    ],
    [
      "26237254577743/1521",
      "64195761425705433328/59319"
    ],
    [
      "34422059675527/1849",
      "60366791367160585904/79507"
    ],
    [
      "1071213886219/49",
      "57609912776124428/343"
    ],
    [
      "334549119776911/14161",
      "917324061339761594224/1685159"
    ],
    [
      "24057082300111/961",
      "27069302511279159776/29791"
    ],
    [
      "1422590310943/49",
      "686717324276509856/343"
    ],
    [
      "28343293535",
      "-1808869545689200"
    ],
    [
      "2795969649596695/94249",
      "-63115493834990247059920/28934443"
    ],
    [
      "1244932703",
      "-4295877525837296"
    ],
    [
      "1588048159279/49",
      "-1020370506585638032/343"
    ],
    [
      "-1523953496064353/173889",
      "-408881406329107132340464/72511713"
    ],
    [
      "28572674647",
      "-1872985304623120"
    ],
    [
      "213705866383207/10201",
      "-234086885442042443408/1030301"
    ],
    [
      "863305992109591/3721",
      "-25054370547651156781744/226981"
    ],
    [
      "2059141531183/49",
      "-2045268646151251984/343"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,1,55,78,86,112,119,124,140,165: 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 >= 18: the 18 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 04:55:04",
  "updated_at": "2026-11-04 04:55:04"
}