{
  "id": 6501,
  "curve_key": "24948640301673187703856:-3938183690475161463906137824118880",
  "ainvs": [
    "0",
    "0",
    "0",
    "-519763339618191410497",
    "4558082975086992435076548407545"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 154.71331882998166,
  "faltings_height": 10.649689280474568,
  "conductor": "128819367123292381683043218556548306026945730239879506708",
  "bad_primes": [
    "2",
    "19",
    "61",
    "218111",
    "333226853",
    "4089136407467",
    "93495040721692249378889723"
  ],
  "discriminant": "11346925133688086148169178863335000988077487702449546468867472",
  "regulator": "447892444475954009.6314067006165744903217618699997598753936319241052744540196",
  "points": [
    [
      "493296602381",
      "-346103618108677727"
    ],
    [
      "26049087967",
      "-2948637055613003"
    ],
    [
      "256451464913/64",
      "815945909691627161/512"
    ],
    [
      "913206824497/81",
      "264064331623726637/729"
    ],
    [
      "11184625633003/961",
      "8700984934956826481/29791"
    ],
    [
      "40312238396835/6241",
      "-597837007676058061415/493039"
    ],
    [
      "43920308379312/3721",
      "-58999755325953665807/226981"
    ],
    [
      "5550889973728/441",
      "-926574458315565599/9261"
    ],
    [
      "538990822117243/36481",
      "2246123019030883664161/6967871"
    ],
    [
      "7478262160554707/552049",
      "-22434455521394824287313/410172407"
    ],
    [
      "8696914104523429/644809",
      "-18840471982491066958229/517781627"
    ],
    [
      "17752093159954624/1697809",
      "-1142240092176314643371731/2212245127"
    ],
    [
      "218422188937/16",
      "5226869351573083/64"
    ],
    [
      "335846010041/25",
      "-833350609224161/125"
    ],
    [
      "118009111962264569/8596624",
      "-2507227572421144377768589/25205301568"
    ],
    [
      "54185125593/4",
      "437353455090101/8"
    ],
    [
      "430074227429815/24649",
      "-3463434686588676029545/3869893"
    ],
    [
      "14700794703",
      "-306899063504741"
    ],
    [
      "34608911057449/2500",
      "15685173411079610707/125000"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,3,12,27,31,79,122,126,191,195: 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 04:54:21",
  "updated_at": "2026-11-04 04:54:21"
}