{
  "id": 6512,
  "curve_key": "65892969046561116681:-552947073097120360132738315365",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-1372770188470023264",
    "639985038650044815789656644"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 136.97011959819213,
  "faltings_height": 9.332712117921304,
  "conductor": "1271186737100143303970841727841690041815960146380218",
  "bad_primes": [
    "2",
    "3",
    "7",
    "71",
    "2389",
    "550573638757",
    "36010374752173820440013854445407"
  ],
  "discriminant": "-11372036550097881997323150097271759114085579469517430228",
  "regulator": "13990975023086335.952064615528307334890283619481792356849779003803748157909",
  "points": [
    [
      "3620426666",
      "-207666623831476"
    ],
    [
      "4941261594410",
      "-10983907201793564278"
    ],
    [
      "-8675076124438/8281",
      "22961655059309733710/753571"
    ],
    [
      "-917389436179/2401",
      "3917435721035836691/117649"
    ],
    [
      "-60051783289/289",
      "148716242093321194/4913"
    ],
    [
      "42207262718/2209",
      "-2572136049949255820/103823"
    ],
    [
      "6263766422078/22201",
      "-54869764191914372882/3307949"
    ],
    [
      "301571383949/400",
      "46337349223387003/8000"
    ],
    [
      "1363644269777/961",
      "1172547344175199445/29791"
    ],
    [
      "3760539086",
      "-220587175570900"
    ],
    [
      "8824615834/9",
      "-415354207750162/27"
    ],
    [
      "1338207221",
      "-34632931968616"
    ],
    [
      "13124013936/25",
      "-1000074593931506/125"
    ],
    [
      "316087991447333/185761",
      "-4550893275517452040163/80062991"
    ],
    [
      "495844142589581/1026169",
      "-9833457189560558480489/1039509197"
    ],
    [
      "-4491426917582/7569",
      "-23240725650877619650/658503"
    ],
    [
      "759418922",
      "-5954139800950"
    ],
    [
      "-129067943605/289",
      "-167617229168030729/4913"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,3,91,98,102,141,149,160,186,198: 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 04:55:40",
  "updated_at": "2026-11-04 04:55:40"
}