{
  "id": 6187,
  "curve_key": "2080062201892708201:-3050678545222223928231788725",
  "ainvs": [
    "1",
    "0",
    "1",
    "-43334629206098088",
    "3530877942247836371982322"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 126.57032310111654,
  "faltings_height": 8.434992310043876,
  "conductor": "65464962044891481098183123461289160971778",
  "bad_primes": [
    "2",
    "3",
    "41",
    "266117731889802768691801314883289272243"
  ],
  "discriminant": "-177615874612311788453316326764004537026481493604908",
  "regulator": "7640017432398204727.076683346374056257275234398494248702716219199834144",
  "points": [
    [
      "-290385916385701/4618201",
      "24324673797296714854053/9924513949"
    ],
    [
      "97475104663/1156",
      "27126163503870965/39304"
    ],
    [
      "416663410274/3025",
      "69630216479870958/166375"
    ],
    [
      "3154297159991/16900",
      "3063570875730301221/2197000"
    ],
    [
      "851202342484/8649",
      "376621079973182185/804357"
    ],
    [
      "3722885494604/7225",
      "6671339518236104622/614125"
    ],
    [
      "15025928222825/135424",
      "14835864346606721331/49836032"
    ],
    [
      "5240285574539/1190281",
      "2373332888383203295377/1298596571"
    ],
    [
      "11386213189451/156025",
      "53622832875791126139/61629875"
    ],
    [
      "44350297717115/413449",
      "90810931720401361827/265847707"
    ],
    [
      "176366988042131/1646089",
      "724856779594131088059/2111932187"
    ],
    [
      "380371811281826/2430481",
      "2890542817985041846845/3789119879"
    ],
    [
      "3940940197927439/17656804",
      "165523403738292642297765/74193890408"
    ],
    [
      "99064472048900969/245862400",
      "27660961781393741827212843/3855122432000"
    ],
    [
      "160837774947784469/176810209",
      "62945773058288842762863465/2351045349073"
    ],
    [
      "4149866728519830917/27986947849",
      "2829912687612546448123857897/4682020466502757"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,5,8,26,33,45,67,74,115,125: 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 14:41:36",
  "updated_at": "2026-10-30 14:41:36"
}