{
  "id": 6182,
  "curve_key": "17096701486954921:-2228335493231872399858325",
  "ainvs": [
    "1",
    "1",
    "0",
    "-356181280978227",
    "2579091857646540795945"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 112.13298583293857,
  "faltings_height": 7.1620735059547815,
  "conductor": "27175971268218999870987545081109578626254",
  "bad_primes": [
    "2",
    "3",
    "113",
    "5309",
    "8293",
    "910397360927738869218077930189"
  ],
  "discriminant": "18425308519852481912529555564992294308600212",
  "regulator": "71058349554506336.42207440464898247966125349215260338499782033408975342",
  "points": [
    [
      "-21592779",
      "14239072299"
    ],
    [
      "-18114294",
      "55572210759"
    ],
    [
      "4811146",
      "31251649255"
    ],
    [
      "11432460",
      "1134172965"
    ],
    [
      "11671511",
      "3438024900"
    ],
    [
      "4483111434",
      "300166496193159"
    ],
    [
      "25656145239",
      "4109468303329509"
    ],
    [
      "5976900097/576",
      "9001593542059/13824"
    ],
    [
      "1553485818/121",
      "14718260146317/1331"
    ],
    [
      "3369785517/289",
      "16463770685811/4913"
    ],
    [
      "3388640693/169",
      "129931346340927/2197"
    ],
    [
      "4190459440/529",
      "194121955607195/12167"
    ],
    [
      "39931153977/1849",
      "5598141846195909/79507"
    ],
    [
      "57745782218/9409",
      "22800745448336023/912673"
    ],
    [
      "171917609191/1369",
      "70516280552401252/50653"
    ],
    [
      "1138198637189/7225",
      "1205912760443315143/614125"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,2,15,24,33,36,40,62,73,87: 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 14:41:01",
  "updated_at": "2026-10-30 14:41:01"
}