{
  "id": 6533,
  "curve_key": "266174273001:-152315099984784501",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-5545297354",
    "176292011306728"
  ],
  "rank_lower_bound": 12,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 79.12945559243752,
  "faltings_height": 4.565863593791333,
  "conductor": "16080655039203067242426890",
  "bad_primes": [
    "2",
    "5",
    "9413",
    "331147",
    "515887318195399"
  ],
  "discriminant": "-2512602349875479256629201562500",
  "regulator": "126002525.10110476865515299368340012793189312594698793697951166",
  "points": [
    [
      "10152774/121",
      "-23176137664/1331"
    ],
    [
      "516008742",
      "-11721822570896"
    ],
    [
      "2300189/16",
      "-3107290965/64"
    ],
    [
      "656307708/1849",
      "-16487398290772/79507"
    ],
    [
      "-162704978/4761",
      "5935966951336/328509"
    ],
    [
      "-347357/4",
      "14148757/8"
    ],
    [
      "1211793/49",
      "-2531264653/343"
    ],
    [
      "96069322/841",
      "782647094956/24389"
    ],
    [
      "130142138/289",
      "1464564524202/4913"
    ],
    [
      "3547483/49",
      "-4272907953/343"
    ],
    [
      "2378923468/15129",
      "-105284096251832/1860867"
    ],
    [
      "241125082/3721",
      "-2149404457476/226981"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,11,80,86,115,120,140,145,170,185: 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 >= 12: the 12 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 04:58:07",
  "updated_at": "2026-11-04 04:58:07"
}