{
  "id": 6071,
  "curve_key": "1114788256500280896:-1256343565390573786826912640",
  "ainvs": [
    "0",
    "0",
    "0",
    "-23224755343755852",
    "1454101348831682623642260"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 124.79600616160778,
  "faltings_height": 8.34972119317277,
  "conductor": "743330050898611727181276787476878581612392",
  "bad_primes": [
    "2",
    "3",
    "7",
    "13",
    "169399",
    "669725691823818819025652417163929"
  ],
  "discriminant": "-111685685052660028964828249430830396158923766149888",
  "regulator": "160249153910427164.5358090414635435076559612608410760214520201878728073",
  "points": [
    [
      "-149463183",
      "1259544692133"
    ],
    [
      "-66012508862199/466489",
      "439974776065486823343/318611987"
    ],
    [
      "-6454524826814/97969",
      "50370264926068216402/30664297"
    ],
    [
      "-2213815514/25",
      "209773800197234/125"
    ],
    [
      "2017822641/16",
      "46634133885417/64"
    ],
    [
      "4107768846/25",
      "180021263406894/125"
    ],
    [
      "17677971114/25",
      "2300145800905638/125"
    ],
    [
      "26046635809/64",
      "3946409908252721/512"
    ],
    [
      "218753894706/2401",
      "36147178936197342/117649"
    ],
    [
      "247874727780/1681",
      "76613295128625450/68921"
    ],
    [
      "1985289026562/51529",
      "9184233373501968522/11697083"
    ],
    [
      "9239470065654/51529",
      "20443416718301696574/11697083"
    ],
    [
      "32545904826426/1681",
      "185665458472758058158/68921"
    ],
    [
      "35890365958548/237169",
      "136906426961342728326/115501303"
    ],
    [
      "57805332433306/36481",
      "437536165503560682338/6967871"
    ],
    [
      "104259853724316/546121",
      "804971618142813905238/403583419"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,7,44,49,64,66,82,101,124,139: 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": "2026-10-28 21:14:25",
  "updated_at": "2026-10-28 21:14:25"
}