{
  "id": 6503,
  "curve_key": "17111257295573655089616:-2234942374031326489238824635437760",
  "ainvs": [
    "0",
    "0",
    "0",
    "-356484526991117814367",
    "2586738858832553806989380365090"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 153.58207056349107,
  "faltings_height": 10.587227537855103,
  "conductor": "273533793796876732889999722911159668751149628173587165378376",
  "bad_primes": [
    "2",
    "419",
    "1013",
    "134764445027",
    "594452802907",
    "381063290769779",
    "2638807524349021"
  ],
  "discriminant": "8753081401500055452479991133157109400036788101554789292108032",
  "regulator": "856708172309491669.5088070501031045033668742197602488434983928708639439596085",
  "points": [
    [
      "402733330137",
      "-255303746030590358"
    ],
    [
      "170345774049/4",
      "-64319365597065209/8"
    ],
    [
      "123654481309/49",
      "447637919646192794/343"
    ],
    [
      "960059166849/169",
      "1896232896078262286/2197"
    ],
    [
      "9934947665829/1225",
      "20518964160928021858/42875"
    ],
    [
      "50752476961830/5329",
      "-91633285874834830400/389017"
    ],
    [
      "560995291753/49",
      "26731079415399206/343"
    ],
    [
      "3659111421521/256",
      "2627589300752223193/4096"
    ],
    [
      "470415904454646/22801",
      "6897760971606782585788/3442951"
    ],
    [
      "51752072150781/3481",
      "-155452506796272758122/205379"
    ],
    [
      "-11981068159",
      "-2266709053565042"
    ],
    [
      "202759121641/25",
      "-59817822928847686/125"
    ],
    [
      "-329199492123/49",
      "-741902518612874578/343"
    ],
    [
      "26130191071",
      "-3333628928233712"
    ],
    [
      "6753348066949/169",
      "-15866028281402067586/2197"
    ],
    [
      "-1061971472931/49",
      "-124961449068042586/343"
    ],
    [
      "-52304369553467/2401",
      "14177816731506002546/117649"
    ],
    [
      "-49308454015288613/2362369",
      "-3509394808249363728747088/3630961153"
    ],
    [
      "19139155423759409/894916",
      "-1844058008543221317839599/846590536"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,4,21,26,35,73,84,96,151,170: 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 >= 19: the 19 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 04:54:35",
  "updated_at": "2026-11-04 04:54:35"
}