{
  "id": 6504,
  "curve_key": "177900719671435204595025:-75328062366908713292414683013403225",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-3706264993154900095730",
    "87185257370033873299694608141297"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 160.61432161675782,
  "faltings_height": 11.210008826483593,
  "conductor": "8228449224205764207124047078328908263712751841550",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "53",
    "67",
    "744843943",
    "356371063635541",
    "395905015583754707"
  ],
  "discriminant": "-25460944043131256715924181305813687745322277173224203249745920000",
  "regulator": "1299989947977981.1644119293745432993604783712476006343664866194920276216312",
  "points": [
    [
      "810839568025/9",
      "594333030616734757/27"
    ],
    [
      "3391339814369",
      "-6244354911331460465"
    ],
    [
      "89680651201489/1521",
      "509024567759157594425/59319"
    ],
    [
      "33659191649",
      "754475085674255"
    ],
    [
      "8798207868181/289",
      "7874074289033508295/4913"
    ],
    [
      "-18819180119879/529",
      "160499781423411939385/12167"
    ],
    [
      "-39643194555599/729",
      "222617808826889319485/19683"
    ],
    [
      "-5192503161329/289",
      "-59764358005205881775/4913"
    ],
    [
      "-149985374718959/7569",
      "-8141145052376267619095/658503"
    ],
    [
      "25992055934844241/734449",
      "369590584321949965531975/629422793"
    ],
    [
      "-68257938215",
      "-4705659831014329"
    ],
    [
      "-59875976815",
      "-9717900796399057"
    ],
    [
      "1079963227835809/36481",
      "12867724998038845474225/6967871"
    ],
    [
      "371393895809",
      "223468678910881615"
    ],
    [
      "42118515169",
      "2408263132791695"
    ],
    [
      "32573900770849/961",
      "21110600480596036625/29791"
    ],
    [
      "-780530582161931/14641",
      "-20450194275295186125925/1771561"
    ],
    [
      "35970298129",
      "640618415063855"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,15,39,67,68,92,108,170,174,195: 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 04:54:43",
  "updated_at": "2026-11-04 04:54:43"
}