{
  "id": 6185,
  "curve_key": "5500364053277961:-428996799285549596335845",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-114590917776624",
    "496524101894893328804"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 108.8314728218046,
  "faltings_height": 7.007203601799688,
  "conductor": "62979444499453069527560665784219464167354",
  "bad_primes": [
    "2",
    "3",
    "3498858027747392751531148099123303564853"
  ],
  "discriminant": "-10202670008911397263464827857043553195111348",
  "regulator": "502583339240875792.7577453340423394722280642591724942385099514353545844",
  "points": [
    [
      "8421358",
      "11342571820"
    ],
    [
      "122539766",
      "1351423465660"
    ],
    [
      "-4092698209024/466489",
      "9161501434369107026/318611987"
    ],
    [
      "-39487196339/46225",
      "242180394258828389/9938375"
    ],
    [
      "-17274995119/1600",
      "1395383627381259/64000"
    ],
    [
      "-747511322/361",
      "184681847063770/6859"
    ],
    [
      "-483568406/169",
      "62181820230460/2197"
    ],
    [
      "526689104/49",
      "7718934438510/343"
    ],
    [
      "12151708069/10000",
      "18948559485974047/1000000"
    ],
    [
      "28520371543/6724",
      "5135376981081895/551368"
    ],
    [
      "38614069105/5329",
      "2655524963692324/389017"
    ],
    [
      "161524335091/6724",
      "59392806101674189/551368"
    ],
    [
      "145480663894/1849",
      "54998582565021178/79507"
    ],
    [
      "588301282640/24649",
      "412370084234122854/3869893"
    ],
    [
      "6791946063046/1729225",
      "23521057077345490324/2273930875"
    ],
    [
      "8237269712866/1104601",
      "8736842763320149174/1160935651"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,4,64,78,83,88,107,116,132,138: 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 14:41:22",
  "updated_at": "2026-10-30 14:41:22"
}