{
  "id": 6526,
  "curve_key": "7915635654101721:-759349593339913458399645",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-164909076127119",
    "878876885370687386409"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 109.97349844302802,
  "faltings_height": 7.121288009582369,
  "conductor": "2592577917287939260041241026684010025843766",
  "bad_primes": [
    "2",
    "3",
    "607",
    "613",
    "2900009921",
    "44492766300765963486234739"
  ],
  "discriminant": "-46666402511182906680742338480312180465187788",
  "regulator": "12556720527367.53768584373724976637665287641542134104378782613441537976",
  "points": [
    [
      "-3357318",
      "37347184335"
    ],
    [
      "11752102818/49",
      "-1272265829293731/343"
    ],
    [
      "-10986308",
      "36945751805"
    ],
    [
      "4561077178/289",
      "-230860271972025/4913"
    ],
    [
      "1538301964/121",
      "38501949587463/1331"
    ],
    [
      "1261182768/169",
      "17543286601905/2197"
    ],
    [
      "25915176978/3721",
      "1873337268464175/226981"
    ],
    [
      "4454334195/121",
      "281366948799402/1331"
    ],
    [
      "1320023181330/26569",
      "1470574995938093049/4330747"
    ],
    [
      "5400634",
      "12071353223"
    ],
    [
      "10343601514/2809",
      "-2669953790618765/148877"
    ],
    [
      "-8975520",
      "40451416077"
    ],
    [
      "1839496561/225",
      "29621841974071/3375"
    ],
    [
      "-9738969",
      "-39507206706"
    ],
    [
      "6715036",
      "8616324285"
    ],
    [
      "15009391186/2025",
      "727347343223231/91125"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,7,12,17,44,52,61,88,163,172: 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); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank >= 16: the 16 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.",
  "created_at": "2026-11-04 04:57:18",
  "updated_at": "2026-11-04 04:57:18"
}