{
  "id": 6510,
  "curve_key": "40356005567017332816:-266224112112225938843347968960",
  "ainvs": [
    "0",
    "0",
    "0",
    "-840750115979527767",
    "308129759389150392179800890"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 135.5082719832644,
  "faltings_height": 9.216797849111105,
  "conductor": "164297669584601186159725572720336551472620693277672",
  "bad_primes": [
    "2",
    "3",
    "7",
    "325987439651986480475645977619715379905993439043"
  ],
  "discriminant": "-2981016916943003921682060791437786389919229858830080768",
  "regulator": "2584668277061101.9918829568474191511945768389806982861730441324810772830084",
  "points": [
    [
      "7881983598",
      "-695237245220196"
    ],
    [
      "1653431526769/576",
      "-2029257369732113801/13824"
    ],
    [
      "-182513919159/784",
      "486541944533544507/21952"
    ],
    [
      "991827785929/3969",
      "2665514276285853826/250047"
    ],
    [
      "609647038905/1369",
      "237786421104407070/50653"
    ],
    [
      "33745228843/169",
      "-26746919628439724/2197"
    ],
    [
      "5628224271219/12769",
      "-6947467493398972596/1442897"
    ],
    [
      "141274960243957/178929",
      "884344715453202126382/75686967"
    ],
    [
      "6855173726142/7921",
      "10661544616265286048/704969"
    ],
    [
      "21962737377373/40401",
      "-27816078168789271046/8120601"
    ],
    [
      "-349169823294/361",
      "100911996829769958/6859"
    ],
    [
      "561052447",
      "-3610104382108"
    ],
    [
      "649684078",
      "-6011036426204"
    ],
    [
      "8395836009/16",
      "-216494047357749/64"
    ],
    [
      "91729055329473/157609",
      "-249901487094571721766/62570773"
    ],
    [
      "-44935970221707/192721",
      "-1875641654014085544174/84604519"
    ],
    [
      "288490737",
      "-9465269676858"
    ],
    [
      "345856044814/841",
      "-137806777361427184/24389"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,4,14,21,37,65,113,141,178,197: 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 >= 18: the 18 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 04:55:26",
  "updated_at": "2026-11-04 04:55:26"
}