{
  "id": 6506,
  "curve_key": "1572604816253609179216:-62047111850149475962191953635264",
  "ainvs": [
    "0",
    "1",
    "0",
    "-32762600338616857900",
    "71813786852678063732257290000"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 146.42106094815352,
  "faltings_height": 10.038180993655944,
  "conductor": "17215711487606289426109405825824671706004755363080",
  "bad_primes": [
    "2",
    "5",
    "23",
    "1838988631",
    "4957400537",
    "2052598913453611940344986917"
  ],
  "discriminant": "22767778442359317766029689204653128331191288967673300000000",
  "regulator": "1243401247499361.3360965767192010687755046253199341981745981184543632551513",
  "points": [
    [
      "13928829414/25",
      "28975461413171562/125"
    ],
    [
      "1118844827680",
      "-1183447824449635180"
    ],
    [
      "-1908865481150/289",
      "-36074121857490350/4913"
    ],
    [
      "8874173264230/1849",
      "-12602579481856246390/79507"
    ],
    [
      "2554210051090/729",
      "-115869127727426230/19683"
    ],
    [
      "2856125490",
      "-39222414922590"
    ],
    [
      "3364010784775/1089",
      "-330869089001057050/35937"
    ],
    [
      "105576360925344/15625",
      "778632851536272062172/1953125"
    ],
    [
      "85662559711489/10000",
      "647888246548153345513/1000000"
    ],
    [
      "200503587943950/78961",
      "1567925191500836745750/22188041"
    ],
    [
      "7842401405/4",
      "983573181329315/8"
    ],
    [
      "427801101550/169",
      "156899590030519250/2197"
    ],
    [
      "27927001210/9",
      "145427567748830/27"
    ],
    [
      "866891115740/289",
      "-112909061967386620/4913"
    ],
    [
      "2674512874",
      "-57624926678330"
    ],
    [
      "-1033461802315/196",
      "858874782275943385/2744"
    ],
    [
      "2988911212150/961",
      "-37051499534564050/29791"
    ],
    [
      "10465929267065/4096",
      "18128462325027999565/262144"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,5,34,43,81,92,99,125,150,169: 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 04:54:57",
  "updated_at": "2026-11-04 04:54:57"
}