{
  "id": 1187,
  "curve_key": "720724161:-21083029079841",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-15015087",
    "24405407799"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    7
  ],
  "naive_height": 61.358979047536515,
  "faltings_height": 3.0762074591528674,
  "conductor": "513110",
  "bad_primes": [
    "2",
    "5",
    "13",
    "3947"
  ],
  "discriminant": "-40577990098780160000000",
  "regulator": "7.25709851041705863257365919696207741747784",
  "points": [
    [
      "-4281",
      "103202"
    ],
    [
      "21407",
      "3073546"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Found by a rational-parameter search of the Kubert-Tate family for torsion 7, at t=-8/5. In y^2+(1-c)xy-by=x^3-bx^2 the parameters are defined by c=t*(t-1);b=t^2*(t-1); Converted to the global minimal model; PARI/GP ellrank returned bounds [2,2]. The 2 supplied points and the torsion subgroup are independently certified by the site.",
  "created_at": "2026-09-23 21:40:04",
  "updated_at": "2026-09-23 21:40:04"
}