{
  "id": 6105,
  "curve_key": "22676079174998447601:-104244532887143488344216031305",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-472418316145800992",
    "120653394663409542397811395"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 133.70352678577714,
  "faltings_height": 9.06371276538763,
  "conductor": "131009313997870661224403129505597972969582",
  "bad_primes": [
    "2",
    "3",
    "7",
    "23",
    "31",
    "27357493",
    "146647847",
    "363488026492723498459"
  ],
  "discriminant": "459039832713041460827865226177104599629400195365896192",
  "regulator": "873094578976225030.3099155916598027850235323704441574393098662481355487",
  "points": [
    [
      "330335567",
      "802058935697"
    ],
    [
      "-253633214582183/1471369",
      "25048457484461470599293/1784770597"
    ],
    [
      "-52496376455/81",
      "9064530712451521/729"
    ],
    [
      "36910258681/81",
      "27933322763429/729"
    ],
    [
      "669054937981/400",
      "506591855955997219/8000"
    ],
    [
      "185749132297/361",
      "25478877413675603/6859"
    ],
    [
      "299196215089/625",
      "32039385976676513/15625"
    ],
    [
      "319026770669/25",
      "179937028409910893/125"
    ],
    [
      "1101316070677/121",
      "1152547348142701747/1331"
    ],
    [
      "2551181719591/46225",
      "96738712619829698791/9938375"
    ],
    [
      "2966259940405/2809",
      "4209004116403803085/148877"
    ],
    [
      "15433821254893/178929",
      "679267459060137483895/75686967"
    ],
    [
      "38927436085177/269361",
      "1040515059416357147243/139798359"
    ],
    [
      "39591044261749/120409",
      "38918574249770272291/41781923"
    ],
    [
      "225056259981253/137641",
      "3114366508373359799659/51064811"
    ],
    [
      "213141828789726535/207849889",
      "80100001563199815227017447/2996571849713"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,7,47,53,75,89,96,107,113,125: 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 02:29:25",
  "updated_at": "2026-10-29 02:29:25"
}