{
  "id": 6181,
  "curve_key": "565805733148261521:-425904781674492609684039225",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-11787619440588782",
    "492945352107197232466685"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 122.63251607189349,
  "faltings_height": 7.980784037217658,
  "conductor": "6872032306498553291336308079119666264818",
  "bad_primes": [
    "2",
    "3",
    "11",
    "34707233871204814602708626662220536691"
  ],
  "discriminant": "-150478375774377025107902066651028422048632946688",
  "regulator": "27503319688014276.13129319976274502113131922213220028379149488541455555",
  "points": [
    [
      "-122913635",
      "291359570055"
    ],
    [
      "32162511",
      "383514471421"
    ],
    [
      "61504857",
      "24727917235"
    ],
    [
      "64759621",
      "34217164491"
    ],
    [
      "670816305",
      "17159188931275"
    ],
    [
      "1986042969",
      "88377533358835"
    ],
    [
      "2694386451",
      "139745635926001"
    ],
    [
      "-45697993731/529",
      "11326790897748257/12167"
    ],
    [
      "-18849254535/289",
      "4874462053318787/4913"
    ],
    [
      "-9270238479/121",
      "1294850027524457/1331"
    ],
    [
      "6352677639/100",
      "22054513038943/1000"
    ],
    [
      "6553622145/121",
      "153976473508745/1331"
    ],
    [
      "11507037201/25",
      "1202716270796711/125"
    ],
    [
      "91039434879/49",
      "27422892236715541/343"
    ],
    [
      "162446852871/2401",
      "8422666172086963/117649"
    ],
    [
      "503926735909/363609",
      "151367893711178245669/219256227"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,3,18,21,30,45,55,65,80,101: 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": "2026-10-30 14:40:54",
  "updated_at": "2026-10-30 14:40:54"
}