{
  "id": 6500,
  "curve_key": "1716984300323236694736:-44622595577496640827033434047680",
  "ainvs": [
    "0",
    "0",
    "0",
    "-35770506256734097807",
    "51646522659139630586844252370"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 146.68456917324357,
  "faltings_height": 10.262176099267712,
  "conductor": "71282932021622334919443458087017223694230637218579805784",
  "bad_primes": [
    "2",
    "19",
    "41",
    "3769",
    "87587",
    "49118132515147",
    "705424396042938613529302168457"
  ],
  "discriminant": "1776940929435001564871886523193165352249781324584757398583552",
  "regulator": "97221898382089779.59550479739059723608661250666460109337700891927176513648260",
  "points": [
    [
      "33318295961",
      "-5987217156573782"
    ],
    [
      "96660952661601/25",
      "-950333606138319520526/125"
    ],
    [
      "-20558953365959/3481",
      "48989501846135497322/205379"
    ],
    [
      "66338265739/2025",
      "20472672507940336112/91125"
    ],
    [
      "1243666497737/5329",
      "80959678279595498758/389017"
    ],
    [
      "37023883349/25",
      "5477331643055618/125"
    ],
    [
      "4329473728933/729",
      "4342711026829763186/19683"
    ],
    [
      "138031490330441/14641",
      "1316652126954688004038/1771561"
    ],
    [
      "60947953323085/4761",
      "427266449148921480010/328509"
    ],
    [
      "4584701863471521/473344",
      "-255149307465193475427343/325660672"
    ],
    [
      "41946011129/4",
      "-7287053903704811/8"
    ],
    [
      "191171018337/16",
      "-73807506046667887/64"
    ],
    [
      "13616284998790/1849",
      "-34435641953511923240/79507"
    ],
    [
      "349790335017/49",
      "-137231257927682918/343"
    ],
    [
      "83292107433/16",
      "5163602889261589/64"
    ],
    [
      "-26875770258314/4225",
      "-40539375198951218816/274625"
    ],
    [
      "27448177541",
      "-4444022449590302"
    ],
    [
      "-1021817659234303/155236",
      "2669446693321859238953/61162984"
    ],
    [
      "1195733058977/49",
      "-1270044458594763118/343"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,4,59,72,73,82,108,121,138,163: 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 04:54:14",
  "updated_at": "2026-11-04 04:54:14"
}