{
  "id": 6444,
  "curve_key": "491175963260730201:-346918700994120639005171805",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-10232832567931879",
    "401526276856958881572289"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 122.22226538654057,
  "faltings_height": 8.03918860133029,
  "conductor": "616078358480899884602577360817997503648061974",
  "bad_primes": [
    "2",
    "13",
    "67",
    "353661514627382252929148886807116821841597"
  ],
  "discriminant": "-1073208500473727598977689762544951651354923958708",
  "regulator": "87551858558439.2824958595419158866684275177416925933519397641455068370509",
  "points": [
    [
      "7279468428/49",
      "-504142050307265/343"
    ],
    [
      "22636829012",
      "-3405810656752487"
    ],
    [
      "-4794029220/121",
      "1148671077130363/1331"
    ],
    [
      "-366587609/81",
      "487802919876211/729"
    ],
    [
      "2235964737/256",
      "2290875305036153/4096"
    ],
    [
      "-2085162514/121",
      "-1007291349023511/1331"
    ],
    [
      "23950060",
      "-412549495237"
    ],
    [
      "1443064032512/22201",
      "347101890113453325/3307949"
    ],
    [
      "98157379403/1444",
      "7765678037434911/54872"
    ],
    [
      "30597769288/441",
      "-1480638891001523/9261"
    ],
    [
      "-114220908",
      "-283061650477"
    ],
    [
      "48518127",
      "-138806446370"
    ],
    [
      "777809917/9",
      "-10890768856190/27"
    ],
    [
      "67216985",
      "-131945439023"
    ],
    [
      "414142133177/5041",
      "-121570709086760905/357911"
    ],
    [
      "415919502520/3721",
      "-183609845641724667/226981"
    ],
    [
      "431295662",
      "-8730394414725"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,7,55,60,64,110,130,149,152,165: 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 17, proved: the 17 listed points come from the point search and are independent, and the 2-descent upper bound (ellrank) is 17. Torsion trivial.",
  "created_at": "2026-11-03 14:18:22",
  "updated_at": "2026-11-03 14:18:22"
}