{
  "id": 6525,
  "curve_key": "10523543227858416:-1095913952458740992032920",
  "ainvs": [
    "0",
    "0",
    "1",
    "-219240483913717",
    "1268418926456876148186"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 110.70726181354092,
  "faltings_height": 7.108251761998061,
  "conductor": "502455899915836093929682048280723478616723",
  "bad_primes": [
    "41",
    "305023",
    "5570536413582967",
    "7212477772603119683"
  ],
  "discriminant": "-20600691896549279851116963979509662623285643",
  "regulator": "40698693770320.54372681211187605240635100373919784395474972654588612469",
  "points": [
    [
      "153352257/49",
      "8491790972603/343"
    ],
    [
      "12741229259",
      "-1438191448961022"
    ],
    [
      "-25786531734/2401",
      "5744644780434911/117649"
    ],
    [
      "63324125129/3025",
      "-12727797567080133/166375"
    ],
    [
      "75929226817/8464",
      "-3781632700400577/778688"
    ],
    [
      "75198352889/10000",
      "-6708029811351213/1000000"
    ],
    [
      "16613123285/1369",
      "1006663006706321/50653"
    ],
    [
      "21567671",
      "81070498565"
    ],
    [
      "410497084209/16384",
      "224926672018733865/2097152"
    ],
    [
      "250725848473/40401",
      "98403508549676546/8120601"
    ],
    [
      "-5270533109/1521",
      "2643869810023246/59319"
    ],
    [
      "257191368097/35344",
      "50765167833430191/6644672"
    ],
    [
      "70307517457/13456",
      "25435134676754535/1560896"
    ],
    [
      "-35824071/4",
      "401084620207/8"
    ],
    [
      "18806633864835/2679769",
      "38102330471351175561/4386781853"
    ],
    [
      "-14112764",
      "39391209794"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,7,49,55,92,100,111,123,128,153: 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 >= 16: the 16 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 04:57:11",
  "updated_at": "2026-11-04 04:57:11"
}