{
  "id": 6485,
  "curve_key": "1780530412216408326984754881:-75539951943981027920800025092429846130721",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-37094383587841840145515727",
    "87430499935172500356082174687385737879"
  ],
  "rank_lower_bound": 20,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 188.25096061577074,
  "faltings_height": 13.526537027338431,
  "conductor": "150042411546129109678533765048810322892047214841958925629770",
  "bad_primes": [
    "2",
    "3",
    "5",
    "19",
    "47",
    "109",
    "510233",
    "33567965926342972952914286179074343117294405193"
  ],
  "discriminant": "-35584006371595405919618132623445045676991719329491311922497843148435200000000",
  "regulator": "1231175078898561316.68919442333532976619392520030638069442882615653995497940727",
  "points": [
    [
      "388789220075581/144",
      "4558710576205440021719/1728"
    ],
    [
      "250021739694887",
      "-3952200724100531937694"
    ],
    [
      "4197673374126481/1600",
      "182927220506578725400109/64000"
    ],
    [
      "53757157894637743/3844",
      "-11438966031880648474168487/238328"
    ],
    [
      "3560959092514907/961",
      "-27568722824053110031454/29791"
    ],
    [
      "3008569351387",
      "-1749718162766684194"
    ],
    [
      "7713853710742027/1521",
      "323539049453825871226514/59319"
    ],
    [
      "390790235470647763/20449",
      "233165601266030246424434842/2924207"
    ],
    [
      "762945802211798203/26569",
      "652504402556978185089136082/4330747"
    ],
    [
      "-338491189940237/49",
      "-1284637531923217347142/343"
    ],
    [
      "160609276670917/49",
      "353041877375548232264/343"
    ],
    [
      "-3104882266103",
      "13140481611453225506"
    ],
    [
      "-6174692877613",
      "-9003076788885515194"
    ],
    [
      "10876617205455103/5041",
      "1494640619514284735268058/357911"
    ],
    [
      "159986874441519217/48841",
      "11162826658999767782747716/10793861"
    ],
    [
      "1809850880664286507/591361",
      "729022025713742718472112654/454756609"
    ],
    [
      "2196019308547",
      "-4069500111108609994"
    ],
    [
      "1651151333299891/529",
      "17451683112656620416874/12167"
    ],
    [
      "601145675252303/169",
      "1537726513690766161082/2197"
    ],
    [
      "88569344990989/25",
      "86576679031388595928/125"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,36,40,62,93,109,117,143,163,199: 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 >= 20: the 20 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 04:52:27",
  "updated_at": "2026-11-04 04:52:27"
}