The Lang-Trotter conjecture on average for genus-$2$ curves with Klein-$4$ reduced automorphism group

For an elliptic curve E over Q without complex multiplication, Lang and Trotter conjecture \[ # \{ p<X E has a supersingular reduction at p \} ~ {c√X}{log X} \] as X → ∞, where $c>0$ is a constant depending only on E. Fourvy and Murty obtained an average estimation related to the Lang-Trotter conjecture, called the Lang-Trotter conjecture on average. We considered the Lang-Trotter conjecture for curves of genus 2, and obtained a similar result to the Lang-Trotter conjecture on average for the family of curves Cλ:y²=x(x-1)(x+1)(x-λ)(x-1/ λ). Such curves are characterized as curves of genus two with reduced automorphism group containing the Klein $4$-group.