{
  "id": 6488,
  "curve_key": "14942906735252712770946550561:-1813688077091264447937837102786507801653809",
  "ainvs": [
    "1",
    "0",
    "0",
    "-311310556984431516061386470",
    "2099176015151900501826081277450638173412"
  ],
  "rank_lower_bound": 20,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 194.622102696636,
  "faltings_height": 14.06893616084823,
  "conductor": "4717721772249207926330637464217482024447812014820011301605990",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "239",
    "1619",
    "37588753211",
    "4826273305783",
    "320035715605441459690577420243"
  ],
  "discriminant": "27282467042020061927279922914105390091247496570795617309974922879330114560000000",
  "regulator": "636538277768158637.064614801752982495143144525870534468678414934350760107505562",
  "points": [
    [
      "82235123943514",
      "-729810322387788534632"
    ],
    [
      "47957538947884",
      "-312199089653860687862"
    ],
    [
      "1600361668734",
      "40063265553494951028"
    ],
    [
      "543432530205148/169",
      "73898330346018982513666/2197"
    ],
    [
      "677088943423324/81",
      "6560429865743784653002/729"
    ],
    [
      "-955307785073732/1849",
      "-3779619589051881678351314/79507"
    ],
    [
      "260221418384939164/22801",
      "19713563875246807984363318/3442951"
    ],
    [
      "2124039388196166/169",
      "28800321086186901689196/2197"
    ],
    [
      "11298922759908024/841",
      "450866496647263888006902/24389"
    ],
    [
      "187405249071957004/17161",
      "-3061639905297371852827762/2248091"
    ],
    [
      "10985616179884",
      "-2241259875581044022"
    ],
    [
      "3222762733249036/11449",
      "-54943799476017006740599426/1225043"
    ],
    [
      "53786750391149116/1369",
      "-11384643768217765102084046/50653"
    ],
    [
      "54986427250945324/5041",
      "404254278547560523224998/357911"
    ],
    [
      "-99216020263316852/5329",
      "-14769863649849656343422150/389017"
    ],
    [
      "8183186881011196/529",
      "-381887981392145401158634/12167"
    ],
    [
      "-108402184934276/25",
      "-7253792038192455613342/125"
    ],
    [
      "16119331689686839/676",
      "-1594879828257209476719907/17576"
    ],
    [
      "21278567494264",
      "-71479997480824267382"
    ],
    [
      "11994959793904",
      "-9531110134813527062"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,4,12,13,27,43,151,156,174,184: 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 04:52:50",
  "updated_at": "2026-11-04 04:52:50"
}