{
  "id": 6527,
  "curve_key": "1034502601380969:-33084223183035792953397",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-21552137528770",
    "38291930368399883296"
  ],
  "rank_lower_bound": 15,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 103.71809138321203,
  "faltings_height": 6.48434292813989,
  "conductor": "756008759365704147346258394890094170",
  "bad_primes": [
    "2",
    "5",
    "31",
    "13049",
    "95873",
    "313534811",
    "6217357192685981"
  ],
  "discriminant": "7265244177504416855997543174893804973700",
  "regulator": "505478798411.21040413718818113797884181143663543324509436352184742917",
  "points": [
    [
      "589310608/81",
      "-11905557556816/729"
    ],
    [
      "1793847",
      "2323557368"
    ],
    [
      "27030397/9",
      "-21875744113/27"
    ],
    [
      "6451780272/2209",
      "52737969675128/103823"
    ],
    [
      "3190671",
      "1415581552"
    ],
    [
      "17683088715/5041",
      "865372061580884/357911"
    ],
    [
      "15949287/4",
      "31733048843/8"
    ],
    [
      "4522157",
      "5768998893"
    ],
    [
      "957231003/169",
      "21735579616765/2197"
    ],
    [
      "539949049/225",
      "-2116497059698/3375"
    ],
    [
      "295079728/121",
      "645517670616/1331"
    ],
    [
      "-5354320",
      "435490824"
    ],
    [
      "-15614093/49",
      "-2304110731252/343"
    ],
    [
      "59805639522/24649",
      "-2065110991860032/3869893"
    ],
    [
      "2469895",
      "-358600824"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,18,24,47,62,71,80,88,104,156: 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 04:57:25",
  "updated_at": "2026-11-04 04:57:25"
}