{
  "id": 6104,
  "curve_key": "206260139713781858481:-3027357423738549762493948596585",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-4297086244037122052",
    "3503885908179074526858758055"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 140.3704857828591,
  "faltings_height": 9.596671099926498,
  "conductor": "112538881684078876199780374481021869696746",
  "bad_primes": [
    "2",
    "3",
    "13",
    "23",
    "199",
    "146439585780487",
    "717542011250683003031"
  ],
  "discriminant": "-225646565432832671762104277315959534337598839140401020928",
  "regulator": "2351837781524903699.814684507551567332458873742400892600828438234404139",
  "points": [
    [
      "6655796045",
      "519373797555789"
    ],
    [
      "265305713589",
      "136648939801259805"
    ],
    [
      "-2645814312630739/3924361",
      "606911779276427574783441/7774159141"
    ],
    [
      "-24403843234411/10201",
      "9907475166618716865/1030301"
    ],
    [
      "-1265698184371/17161",
      "138953448383247577935/2248091"
    ],
    [
      "-16577132419/25",
      "9732135475169481/125"
    ],
    [
      "12333206659/9",
      "370881744091319/27"
    ],
    [
      "74325903697/81",
      "13312306040479345/729"
    ],
    [
      "369096220197/289",
      "48889677574690221/4913"
    ],
    [
      "650035877429/625",
      "197461103346538917/15625"
    ],
    [
      "9264923228597/5041",
      "15245799269102256315/357911"
    ],
    [
      "237340164570629/172225",
      "1008910650753825453987/71473375"
    ],
    [
      "3143194606962173/2699449",
      "39440188076385849213159/4435194707"
    ],
    [
      "3916232655242573/96721",
      "244761571291369818447387/30080231"
    ],
    [
      "8583843193242437/1560001",
      "745659033890418610636893/1948441249"
    ],
    [
      "1387773915783354916829/8106063823225",
      "1215367462756287623051087628603303/23078895902061245875"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,7,18,22,27,61,75,99,109,120: 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); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.",
  "created_at": "2026-10-29 02:29:18",
  "updated_at": "2026-10-29 02:29:18"
}