{
  "id": 6532,
  "curve_key": "89835990668697:-880303022739077604669",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-1871583138931",
    "1018869707177124553"
  ],
  "rank_lower_bound": 15,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 96.45359573222791,
  "faltings_height": 5.956436638339101,
  "conductor": "14442008176942461192298997907977036098",
  "bad_primes": [
    "2",
    "248293",
    "1476857",
    "19692219554140193882708149"
  ],
  "discriminant": "-28884016353884922384597995815954072196",
  "regulator": "544454148391.11746080890410509625535278517552860780833579594502327720",
  "points": [
    [
      "6541198/9",
      "-5577726515/27"
    ],
    [
      "79181992",
      "-704530265475"
    ],
    [
      "505039",
      "449708039"
    ],
    [
      "15410909171/24025",
      "1066917002545566/3723875"
    ],
    [
      "139538257/169",
      "418506952538/2197"
    ],
    [
      "-161342321/121",
      "-1422526753091/1331"
    ],
    [
      "-7134043602/7921",
      "-990140188943209/704969"
    ],
    [
      "103924478889/61504",
      "24961156507974887/15252992"
    ],
    [
      "3213738766/1849",
      "138022059561089/79507"
    ],
    [
      "5490300846/51529",
      "-10597084404985763/11697083"
    ],
    [
      "1339079484/3481",
      "-122551131144337/205379"
    ],
    [
      "-17230356/25",
      "-175909826413/125"
    ],
    [
      "188843758149/190096",
      "-31050766208677691/82881856"
    ],
    [
      "1750315579/6561",
      "-390078681369301/531441"
    ],
    [
      "-177429446519/112896",
      "10651650610188883/37933056"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Mestre ten-root construction, roots 0,9,30,31,39,66,89,124,129,149: 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 >= 15: the 15 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 04:58:00",
  "updated_at": "2026-11-04 04:58:00"
}