{
  "id": 6473,
  "curve_key": "-270381980125675719:-3564392318454277209573138285",
  "ainvs": [
    "1",
    "-1",
    "0",
    "5632957919284911",
    "4125454070876803586814349"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 126.8815821850387,
  "faltings_height": 8.631486284790002,
  "conductor": "3787966790370899764343282737908640664928080965962",
  "bad_primes": [
    "2",
    "3",
    "67",
    "25969",
    "57993850631",
    "12256250076377",
    "170162582965031809"
  ],
  "discriminant": "-7363807440481029141883341642494397452620189397830128",
  "regulator": "8935850023230442.80460126410040034019593300494902312784178558109776003937",
  "points": [
    [
      "8871417898/81",
      "-1794058503190237/729"
    ],
    [
      "46138117578",
      "-9910404978455125"
    ],
    [
      "168056104174/729",
      "-82753460014835215/19683"
    ],
    [
      "689212020422/961",
      "-578484153749781607/29791"
    ],
    [
      "381753413/16",
      "132301910670815/64"
    ],
    [
      "-33328445186/225",
      "683281400086357/3375"
    ],
    [
      "-226035677981/1521",
      "-4705632302138708/59319"
    ],
    [
      "3500531456726/10609",
      "7073705434791822881/1092727"
    ],
    [
      "1835477302423/3969",
      "2569862608286609035/250047"
    ],
    [
      "-82325875",
      "1761786935543"
    ],
    [
      "-37616230",
      "1964792888627"
    ],
    [
      "183694958",
      "3370186011047"
    ],
    [
      "216038183",
      "3927414020645"
    ],
    [
      "308131082",
      "5925770013083"
    ],
    [
      "639806723",
      "16420277864372"
    ],
    [
      "53686723121",
      "12439411597515875"
    ],
    [
      "-1920419926/49",
      "672539135580605/343"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,9,27,31,64,75,78,103,126,137: 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 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-11-04 02:10:54",
  "updated_at": "2026-11-04 02:10:54"
}