{
  "id": 6132,
  "curve_key": "10305152871754825:-1066627837248037979373125",
  "ainvs": [
    "1",
    "1",
    "0",
    "-214690684828225",
    "1234522870323369723625"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 110.65308869950422,
  "faltings_height": 7.115118286071306,
  "conductor": "6686283657824244240901969615219363968150",
  "bad_primes": [
    "2",
    "3",
    "5",
    "44575224385494961606013130768129093121"
  ],
  "discriminant": "-25073563716840915903382386057072614880562500",
  "regulator": "1764317418809131.181308817390080488457486583816451509908691478157267111",
  "points": [
    [
      "-16796070",
      "10116702785"
    ],
    [
      "-13715494",
      "39994666241"
    ],
    [
      "-2145995",
      "41054271085"
    ],
    [
      "4340890",
      "19603212805"
    ],
    [
      "21448315/4",
      "123267983585/8"
    ],
    [
      "6365485",
      "11214580645"
    ],
    [
      "34129295/4",
      "39051730555/8"
    ],
    [
      "9215344",
      "6213520051"
    ],
    [
      "12505880",
      "22477289285"
    ],
    [
      "26817140",
      "121489353305"
    ],
    [
      "27937855",
      "130533668935"
    ],
    [
      "79519610",
      "697809936245"
    ],
    [
      "1675343215/4",
      "68530327392035/8"
    ],
    [
      "-12901884320/961",
      "1227427767153235/29791"
    ],
    [
      "1327379460/169",
      "12575677688335/2197"
    ],
    [
      "1650956365/121",
      "38689047227255/1331"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,4,22,53,58,70,101,121,133,138: 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 12:56:12",
  "updated_at": "2026-10-29 12:56:12"
}