{
  "id": 6522,
  "curve_key": "155446813589318121:-63408708057886251357132405",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-3238475283110794",
    "73389709209950130218824"
  ],
  "rank_lower_bound": 16,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 118.82328686987849,
  "faltings_height": 7.821588894284115,
  "conductor": "13621931438642364903907665836794048470758",
  "bad_primes": [
    "2",
    "19",
    "79",
    "197",
    "17961409974941",
    "67494401272556246533"
  ],
  "discriminant": "-153062641903264792249649832467814610524993676388",
  "regulator": "2231281655089.847387732681036593542642832193575814982574569877196044547",
  "points": [
    [
      "10467820582/81",
      "-981678447281078/729"
    ],
    [
      "281764196397/16",
      "-149564347892373725/64"
    ],
    [
      "-245015759316/3721",
      "7653721482813028/226981"
    ],
    [
      "143283525369/7225",
      "79983822220014802/614125"
    ],
    [
      "228960093286/9025",
      "74532746567380224/857375"
    ],
    [
      "28946249966/841",
      "1266788203839666/24389"
    ],
    [
      "820884878017/17161",
      "375654880484247063/2248091"
    ],
    [
      "5540921154757/85264",
      "9227139614392846301/24897088"
    ],
    [
      "1604449485150/22801",
      "1516095292523466568/3442951"
    ],
    [
      "1434972180778/25281",
      "-1082021387587939642/4019679"
    ],
    [
      "759854011/9",
      "-17115517650589/27"
    ],
    [
      "79099378",
      "-558725794230"
    ],
    [
      "209496169774451/5067001",
      "-1150519385961527824220/11405819251"
    ],
    [
      "26924165925749/2053489",
      "-536055575442114967762/2942649737"
    ],
    [
      "19919838312443/458329",
      "-37673471125679482300/310288733"
    ],
    [
      "28588210186806/654481",
      "-65448038309523364630/529475129"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,6,61,85,95,116,131,146,151,195: 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 >= 16: the 16 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 04:56:50",
  "updated_at": "2026-11-04 04:56:50"
}