{
  "id": 6125,
  "curve_key": "38957841184190217681:-240086090393259027069273708825",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-811621691337296202",
    "277877419639547815275613065"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 135.3270352201455,
  "faltings_height": 9.152713787709224,
  "conductor": "20204071657297644439135497567378652495054663514",
  "bad_primes": [
    "2",
    "7",
    "53",
    "256496329586693513",
    "15165467407429906830935137"
  ],
  "discriminant": "859668055556128483056597191597790994917899651413737472",
  "regulator": "2513625068873262007.82174940310712273154958292365100082344025551564138208",
  "points": [
    [
      "-1038755439",
      "351782683127"
    ],
    [
      "408578865",
      "3804102867735"
    ],
    [
      "697603041",
      "7153404138071"
    ],
    [
      "9284494673",
      "890547386970487"
    ],
    [
      "-6556564485959/6889",
      "7844971374354549405/571787"
    ],
    [
      "-36013821951/289",
      "95403979917150631/4913"
    ],
    [
      "8332468999/25",
      "832805963277083/125"
    ],
    [
      "82219487113/121",
      "8430077923496117/1331"
    ],
    [
      "398890529265/6889",
      "8691834225493773765/571787"
    ],
    [
      "1702212475545/3721",
      "346067456890381875/226981"
    ],
    [
      "3239517185673/5041",
      "1666984150465109377/357911"
    ],
    [
      "4449981248227/9801",
      "1672352274092615495/970299"
    ],
    [
      "4628496385825/7569",
      "2106235747681769345/658503"
    ],
    [
      "5650458085519/9801",
      "1220147655445185245/970299"
    ],
    [
      "6973117140985/6889",
      "12700922523626252445/571787"
    ],
    [
      "28283283107129/42025",
      "52033599040002469507/8615125"
    ],
    [
      "601601387285385/76729",
      "14662284034448598968555/21253933"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,3,24,32,57,72,89,102,103,118: 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 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-29 03:14:38",
  "updated_at": "2026-10-29 03:14:38"
}