{
  "id": 6511,
  "curve_key": "920644284723744217:-49624660103840498298195624221",
  "ainvs": [
    "1",
    "1",
    "0",
    "-19180089265078004",
    "57435949186268132132832804"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 132.1485708010605,
  "faltings_height": 9.070100930527106,
  "conductor": "657741528725712916243850244735198959476234712957286",
  "bad_primes": [
    "2",
    "3",
    "19",
    "42461",
    "135881456693947204440617798032043224696229959"
  ],
  "discriminant": "-1424668151219894176584179630096440946225524388265481476",
  "regulator": "5457686906867988.9296734238659049444739599087272892279082832287728452851958",
  "points": [
    [
      "384779894806",
      "238681181716151268"
    ],
    [
      "30464566831",
      "-5317282184604007"
    ],
    [
      "518262084741/16",
      "373094929925221839/64"
    ],
    [
      "9281672798214/13225",
      "-30022649867026322868/1520875"
    ],
    [
      "59921869651/100",
      "-16158893981503731/1000"
    ],
    [
      "45631548950/121",
      "-13563178024663262/1331"
    ],
    [
      "502583160359/2209",
      "-836088205302534377/103823"
    ],
    [
      "799630714/169",
      "-16637157238122078/2197"
    ],
    [
      "-9318903520209/32761",
      "-37444008776920458459/5929741"
    ],
    [
      "345686865163/81",
      "203213597382570241/729"
    ],
    [
      "12506047087/9",
      "1406538213534083/27"
    ],
    [
      "2342664186385/31329",
      "41651781369836260894/5545233"
    ],
    [
      "514804750",
      "13564311400452"
    ],
    [
      "2756492267410182/1394761",
      "144903538601333946703476/1647212741"
    ],
    [
      "9667954875/4",
      "950972170764741/8"
    ],
    [
      "580148312810",
      "-441884556026130022"
    ],
    [
      "1106479994029490/12769",
      "36805614733909046797714/1442897"
    ],
    [
      "5739620803452119/10609",
      "-434835590048222841177409/1092727"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,3,16,115,120,132,141,156,181,196: 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 >= 18: the 18 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 04:55:33",
  "updated_at": "2026-11-04 04:55:33"
}