{
  "id": 6516,
  "curve_key": "572762123285829561:-431273463013639721468211885",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-11932544235121449",
    "499159103693367217887089"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 122.66774064846675,
  "faltings_height": 8.0587538090447,
  "conductor": "6792655770912057655076591471745629739000288246",
  "bad_primes": [
    "2",
    "3",
    "164877103000059467",
    "2288794248468410813834493641"
  ],
  "discriminant": "1100410234887753340122407818422792017718046695852",
  "regulator": "175770043061114.791572134034710919373187492370738094298254425579762473616",
  "points": [
    [
      "5335052857/64",
      "-148168554696505/512"
    ],
    [
      "1721727189011/196",
      "-2258995687151090145/2744"
    ],
    [
      "59559476356/625",
      "-7452358100669321/15625"
    ],
    [
      "627312504469/3721",
      "-411036603893423672/226981"
    ],
    [
      "69204017428/1681",
      "19208798869985735/68921"
    ],
    [
      "124874228066/3025",
      "-46147041048570801/166375"
    ],
    [
      "17673507877/169",
      "1380642111198280/2197"
    ],
    [
      "3562097846287/16129",
      "6019309788423564535/2048383"
    ],
    [
      "403088612668127/1002001",
      "7820733705962760293985/1003003001"
    ],
    [
      "21639586498/169",
      "-2273390267513279/2197"
    ],
    [
      "271126231/4",
      "-336286080781/8"
    ],
    [
      "70823987",
      "-96495201320"
    ],
    [
      "-31237201511/5329",
      "-293417531375800937/389017"
    ],
    [
      "21914990029913/305809",
      "18570436756904634615/169112377"
    ],
    [
      "-20679710765/169",
      "783342731339416/2197"
    ],
    [
      "40874419399613/597529",
      "25277381379659991176/461889917"
    ],
    [
      "17687880271/625",
      "-6704920978591256/15625"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,32,50,61,82,110,117,127,143,178: 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 >= 17: the 17 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 04:56:07",
  "updated_at": "2026-11-04 04:56:07"
}