{
  "id": 6514,
  "curve_key": "122850578261748790921:-1366375626701778278997804732965",
  "ainvs": [
    "1",
    "1",
    "0",
    "-2559387047119766477",
    "1581453270579165442169778705"
  ],
  "rank_lower_bound": 18,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 138.77942899222657,
  "faltings_height": 9.385762176240663,
  "conductor": "1243144522086178951340191789390745525222020024679758998",
  "bad_primes": [
    "2",
    "3",
    "109717225376844435481919",
    "1888406792729165666167018365607"
  ],
  "discriminant": "-7458867132517073708041150736344473151332120148078553988",
  "regulator": "24010083143959589.294498412294610995943716928868549998921907291829728142552",
  "points": [
    [
      "179894196409/64",
      "65956954023968113/512"
    ],
    [
      "58810105723",
      "-14256725145775833"
    ],
    [
      "73698579677/49",
      "11552426835432230/343"
    ],
    [
      "39418001331/25",
      "-4785844263230661/125"
    ],
    [
      "7630826188708/6889",
      "-5874658348288821603/571787"
    ],
    [
      "7810928815789/7225",
      "-5426310653976959758/614125"
    ],
    [
      "157445938939/169",
      "-5221114957508997/2197"
    ],
    [
      "10895370195991/11881",
      "-3062192139642568834/1295029"
    ],
    [
      "2063990254034/2401",
      "-478812976302279739/117649"
    ],
    [
      "530813334544/1369",
      "1288782541340945915/50653"
    ],
    [
      "182300819",
      "33480252727967"
    ],
    [
      "-940010167",
      "56184917731205"
    ],
    [
      "1953897067/4",
      "169290596195661/8"
    ],
    [
      "-1784045067215/1024",
      "-898663151289118695/32768"
    ],
    [
      "21594051101711461/23049601",
      "269960205803936910560385/110661134401"
    ],
    [
      "-1268746909",
      "52786447129775"
    ],
    [
      "-409339780841/225",
      "49630555920193322/3375"
    ],
    [
      "-1846817112",
      "3029483903697"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,4,21,60,83,85,104,109,147,173: 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 >= 18: the 18 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 04:55:53",
  "updated_at": "2026-11-04 04:55:53"
}