{
  "id": 6513,
  "curve_key": "80100001720634699856:-816027572707710294393569603520",
  "ainvs": [
    "0",
    "0",
    "0",
    "-1668750035846556247",
    "944476357300590618511075930"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 137.7484913106309,
  "faltings_height": 9.461104587944583,
  "conductor": "16457814113570642745548909115192863789336093133235112",
  "bad_primes": [
    "2",
    "167",
    "1151",
    "10702626532493641786073103000250279494878245117"
  ],
  "discriminant": "-87950558622921514832213370311590664090212081704008438528",
  "regulator": "8326811878457588.6124603595615606019541663616734195792091092565318091516167",
  "points": [
    [
      "-368531598",
      "-38851141001962"
    ],
    [
      "135232794709",
      "-49728264461906806"
    ],
    [
      "-9094395893/81",
      "-24510259289203748/729"
    ],
    [
      "16593077562533/3721",
      "-65068028665245088474/226981"
    ],
    [
      "14945740518963/5929",
      "-51561764502528367396/456533"
    ],
    [
      "50366415217/64",
      "-5576072440430473/512"
    ],
    [
      "174018280097/289",
      "61750571479591198/4913"
    ],
    [
      "13753557056165/19321",
      "29085816658297382330/2685619"
    ],
    [
      "2241877944549/2209",
      "1786865984395201574/103823"
    ],
    [
      "-1472356933",
      "14479131232012"
    ],
    [
      "-997308503",
      "40209299116762"
    ],
    [
      "2694725677/9",
      "-586385906237342/27"
    ],
    [
      "36413534147/121",
      "-28841258822331572/1331"
    ],
    [
      "-660211813",
      "-41933661393212"
    ],
    [
      "-29462477714187/22201",
      "94833183650937644746/3307949"
    ],
    [
      "-7515303781425485/5517801",
      "340639138875553911489340/12961314549"
    ],
    [
      "-2148469115282/1521",
      "1304029283660326124/59319"
    ],
    [
      "6483259333/9",
      "-291028806684026/27"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,12,27,61,77,96,136,139,188,200: 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 04:55:47",
  "updated_at": "2026-11-04 04:55:47"
}