{
  "id": 6530,
  "curve_key": "1210122884512712001:-1333082195983934603119975905",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-25210893427348167",
    "1542919214617462480818495"
  ],
  "rank_lower_bound": 15,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 124.91458242507662,
  "faltings_height": 8.195653716894066,
  "conductor": "1894719850308446189709956049173645986",
  "bad_primes": [
    "2",
    "17",
    "29",
    "61",
    "97",
    "2423",
    "134033422325366541651733711"
  ],
  "discriminant": "-2897766659211978352636884838975803703420186001408",
  "regulator": "56573354508.852137861202534108449428333825943006757936437807163863985",
  "points": [
    [
      "1772000515",
      "-74303817974706"
    ],
    [
      "3745184879",
      "-228996678775150"
    ],
    [
      "-6984180733/49",
      "513449225907346/343"
    ],
    [
      "166855681/25",
      "146572585599714/125"
    ],
    [
      "8373278523/169",
      "1416023509900134/2197"
    ],
    [
      "2260966579/25",
      "6357404891798/125"
    ],
    [
      "1163729091179/11449",
      "213959179054023338/1225043"
    ],
    [
      "82989668719/529",
      "14644364235229682/12167"
    ],
    [
      "3350393388395/20449",
      "3934413261720349370/2924207"
    ],
    [
      "-53379192309/5041",
      "-481343860913976614/357911"
    ],
    [
      "78980961",
      "-210802955254"
    ],
    [
      "260866189405/2809",
      "-7555442271476590/148877"
    ],
    [
      "100233004590401/625681",
      "-629071401098040777764/494913671"
    ],
    [
      "219579150355/2209",
      "-14343756026370798/103823"
    ],
    [
      "87449443",
      "-83721295538"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,5,35,50,65,100,107,138,143,187: 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 04:57:46",
  "updated_at": "2026-11-04 04:57:46"
}