{
  "id": 6048,
  "curve_key": "-64515344:-889580763800",
  "ainvs": [
    "0",
    "-1",
    "1",
    "1344070",
    "1029159342"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    5
  ],
  "family": null,
  "families": [],
  "naive_height": 55.02803227356536,
  "faltings_height": 2.672864323412955,
  "conductor": "31962203",
  "bad_primes": [
    "7",
    "13",
    "23",
    "15271"
  ],
  "discriminant": "-613357413490695585803",
  "regulator": "36.045896742080231212367678143541155302123932",
  "points": [
    [
      "-585",
      "6506"
    ],
    [
      "-375",
      "21731"
    ],
    [
      "143",
      "34989"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Obtained in my own rational-parameter search of the Tate X_1(5) family, following the construction described by David Renshaw on curves #1185 and #1186. The family is y^2+(1-t)*x*y-t*y=x^3-t*x^2, specialized at t=91/23. Clearing denominators with x=X/23^2 and y=Y/23^3 gives [-68,-2093,-48139,0,0]; the displayed equation is its global minimal model, with discriminant -613357413490695585803. All three listed rational points were checked exactly and returned as an independent set, with rank bounds [3,3]. The rational torsion subgroup is Z/5Z.",
  "created_at": "2026-10-28 17:05:20",
  "updated_at": "2026-10-28 17:05:20"
}