{
  "id": 6103,
  "curve_key": "8766410264544400:-988729984349252296926400",
  "ainvs": [
    "0",
    "1",
    "0",
    "-182633547178008",
    "1144363346933785543588"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 110.5014164567261,
  "faltings_height": 7.206814463700357,
  "conductor": "109913484105165265234727983275144245225200",
  "bad_primes": [
    "2",
    "5",
    "8807",
    "9371",
    "1322831",
    "1455911",
    "25277159",
    "68392676141"
  ],
  "discriminant": "-175861574568264424375564773240230792360320000",
  "regulator": "250648230537432586.7896653874911056900653482662520482446361249663301280",
  "points": [
    [
      "-15792558",
      "9480154048"
    ],
    [
      "-6421322",
      "45302752860"
    ],
    [
      "3197238",
      "24354130340"
    ],
    [
      "14845542",
      "41290196348"
    ],
    [
      "55275414",
      "399920167204"
    ],
    [
      "5726286038",
      "433319615290660"
    ],
    [
      "-1863876979198/120409",
      "676679249029797280/41781923"
    ],
    [
      "-843073424907/111556",
      "1704590782727307815/37259704"
    ],
    [
      "-4777996982/1681",
      "2791521063337440/68921"
    ],
    [
      "-48035483/4",
      "320574599745/8"
    ],
    [
      "25519421/4",
      "123641854237/8"
    ],
    [
      "104382726/25",
      "2665171443476/125"
    ],
    [
      "16656870022/1369",
      "1362425379664020/50653"
    ],
    [
      "1441547143648/398161",
      "5787211693031655690/251239591"
    ],
    [
      "3606765722182/15129",
      "6839059693981126420/1860867"
    ],
    [
      "25654845628854082/1481711049",
      "3212666617494838446707120/57035503409157"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,2,40,44,58,79,84,99,133,137: 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); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-29 02:29:10",
  "updated_at": "2026-10-29 02:29:10"
}