{
  "id": 6478,
  "curve_key": "2062627143643521:-77417954926710599057985",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-42971398825907",
    "89604125241357344275"
  ],
  "rank_lower_bound": 15,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 105.78827063292091,
  "faltings_height": 6.817240675738974,
  "conductor": "2426060742452274879046419019933625406",
  "bad_primes": [
    "2",
    "3",
    "1785018071362301",
    "75506883947453042867"
  ],
  "discriminant": "1609817457775691900541009433514997005402112",
  "regulator": "1282143154747.8233904051598443171570147263511141268109598623709656983",
  "points": [
    [
      "-112983581/25",
      "-1729518154982/125"
    ],
    [
      "1481930107",
      "-57048335778910"
    ],
    [
      "-2705711",
      "-13639180210"
    ],
    [
      "24798433147/1369",
      "-3672358884877414/50653"
    ],
    [
      "126113851/25",
      "-137472811814/125"
    ],
    [
      "87871",
      "9264342926"
    ],
    [
      "-1760853701/289",
      "54995335583458/4913"
    ],
    [
      "21983236883/841",
      "3163356875370218/24389"
    ],
    [
      "50103299267/1369",
      "11043128416548122/50653"
    ],
    [
      "-663461",
      "10854912770"
    ],
    [
      "2364835",
      "1098391034"
    ],
    [
      "2407927",
      "304673330"
    ],
    [
      "5256139",
      "2989327250"
    ],
    [
      "10029835",
      "25832656034"
    ],
    [
      "16431667",
      "61798394210"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,10,24,37,45,48,68,87,111,122: 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); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 15, proved: 15 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-11-04 02:11:31",
  "updated_at": "2026-11-04 02:11:31"
}