{
  "id": 6498,
  "curve_key": "2342644851393916402042027201:-114567780704994895252343408185864127680801",
  "ainvs": [
    "1",
    "0",
    "0",
    "-48805101070706591709208900",
    "132601598038184469079937867258404010000"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 189.08397049167777,
  "faltings_height": 13.623363103056382,
  "conductor": "7225366625201708988410703605170496823734687573654248410",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "31",
    "67",
    "149",
    "449",
    "23189",
    "184411229",
    "1005439531",
    "5235443337652625965613"
  ],
  "discriminant": "-155889355157290562131122737946303661724703221986701909423850322980649356800000",
  "regulator": "25682404972292939.33811618219981944748013213300988900516698180816468098125491",
  "points": [
    [
      "-312958118671210/169",
      "32336204006067627732650/2197"
    ],
    [
      "83795781769532",
      "-764483233120287102184"
    ],
    [
      "-21876999412225/4",
      "122879837730714792025/8"
    ],
    [
      "2366748420658960/441",
      "-46536877260070251061300/9261"
    ],
    [
      "4488435974906",
      "-1991814452424026338"
    ],
    [
      "685555467447278/169",
      "-2575480785680052089806/2197"
    ],
    [
      "38534308730713490/6889",
      "3364319916397017551585650/571787"
    ],
    [
      "2831053621404410/361",
      "104510433786803041068410/6859"
    ],
    [
      "156948811739235482/14161",
      "52024941755199124859347474/1685159"
    ],
    [
      "2263193364730",
      "5808474881198401000"
    ],
    [
      "137776938659180/49",
      "1439078403093216624500/343"
    ],
    [
      "11975131350115/4",
      "-29199692323842709525/8"
    ],
    [
      "45335756621000/169",
      "24019600783852177774700/2197"
    ],
    [
      "532403118815788604/133225",
      "57214078086142392123331228/48627125"
    ],
    [
      "-72956070547465150/10201",
      "11088942529744260106414600/1030301"
    ],
    [
      "800062944126800/18769",
      "-29376655192579163522110700/2571353"
    ],
    [
      "44678496787073/64",
      "5091012804828694543699/512"
    ],
    [
      "364157007744620/169",
      "13443469016682631217000/2197"
    ],
    [
      "-4156531450936540/529",
      "67727096325179940838900/12167"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,13,20,42,55,65,83,95,119,144: 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 04:54:00",
  "updated_at": "2026-11-04 04:54:00"
}