{
  "id": 6497,
  "curve_key": "19567361509653340507736721:-87080408947728466897463108927433671481",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-407653364784444593911182",
    "100787510356269120398445079814488989"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 174.71979056074238,
  "faltings_height": 12.403429536918523,
  "conductor": "2911330511056578480488235124713909835325639289152655130",
  "bad_primes": [
    "2",
    "5",
    "7",
    "37",
    "43",
    "6512969149539799",
    "4013694110277216173007960619324351"
  ],
  "discriminant": "-52670270074838607435721363192930238258297688792441982494725338880000000",
  "regulator": "13804547861455468.57584723328821560715449051374838870820392343068363878612422",
  "points": [
    [
      "198913717017",
      "166042115293260051"
    ],
    [
      "48243265631177",
      "-335055873723976631549"
    ],
    [
      "-613641417703",
      "-346222555694247149"
    ],
    [
      "367814909547",
      "-24645621780256149"
    ],
    [
      "2253591710965277/6241",
      "-12752146285961467901111/493039"
    ],
    [
      "260915446403473/729",
      "-532697368915241181287/19683"
    ],
    [
      "283657637188363/529",
      "2320470872889889943117/12167"
    ],
    [
      "547609739949",
      "204371469938180763"
    ],
    [
      "1221819352301433/2209",
      "21907245380503595901853/103823"
    ],
    [
      "355811206297",
      "28038154160000851"
    ],
    [
      "15469622473803/49",
      "20451891338141034143/343"
    ],
    [
      "114773846297",
      "235608698200592851"
    ],
    [
      "251364977697",
      "119163898433528451"
    ],
    [
      "-445688158453",
      "440390064382086601"
    ],
    [
      "561533554045364763/1630729",
      "73456737268450746093662983/2082440933"
    ],
    [
      "-32397090263",
      "337580066186632051"
    ],
    [
      "377486094777",
      "26347458426958451"
    ],
    [
      "9039248838928617/28561",
      "284367185760379609494459/4826809"
    ],
    [
      "24015087168833737/83521",
      "2068726748237132982603219/24137569"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,5,37,74,79,81,138,140,141,187: 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 >= 19: the 19 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 04:53:53",
  "updated_at": "2026-11-04 04:53:53"
}