{
  "id": 6531,
  "curve_key": "2522657350842321:-33480831706080420177945",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-52555361475882",
    "38750975761618633065"
  ],
  "rank_lower_bound": 15,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 106.39226773541225,
  "faltings_height": 6.929878022817615,
  "conductor": "2179515231804095837331261853836436454",
  "bad_primes": [
    "2",
    "11",
    "985007",
    "47268259",
    "193435250015794738199"
  ],
  "discriminant": "8641620969006550100118165399607994316685312",
  "regulator": "140173252576.17606089551415145304482487511042488733896808963451669984",
  "points": [
    [
      "-21190847/49",
      "2687726252625/343"
    ],
    [
      "624151905",
      "-15592468681561"
    ],
    [
      "-3209489",
      "13206393847"
    ],
    [
      "5946285449/361",
      "-413997503969875/6859"
    ],
    [
      "21221643041/1681",
      "2566626989878687/68921"
    ],
    [
      "27410153/169",
      "-12079937658365/2197"
    ],
    [
      "34799559337/77841",
      "-85078564732979495/21717639"
    ],
    [
      "5113584103/121",
      "360315775855947/1331"
    ],
    [
      "2073226686649/38025",
      "2958830143910518253/7414875"
    ],
    [
      "18597709/25",
      "32134926563/125"
    ],
    [
      "134225",
      "5630132759"
    ],
    [
      "-6117943",
      "-11455203505"
    ],
    [
      "264295369/441",
      "25307585787595/9261"
    ],
    [
      "-5507009495/729",
      "42643871507557/19683"
    ],
    [
      "-10307925751/32761",
      "-44077370504643605/5929741"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,7,16,19,41,52,62,88,130,141: 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 04:57:53",
  "updated_at": "2026-11-04 04:57:53"
}