The average exponent of elliptic curves modulo $p$

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. For a prime $p$ of good reduction for $E$, denote by $e_p$ the exponent of the reduction of $E$ modulo $p$. Under GRH, we prove that there is a constant $C_E\in (0,1)$ such that $$\frac{1}{\pi (x)}\sum _{p\le x}{e}_p=1\mathrm{/}2C_{E}x+O_E(x^{5\mathrm{/}6}(\log x{)}^{4\mathrm{/}3})$$ for all $x\ge 2$, where the implied constant depends on $E$ at most. When $E$ has complex multiplication, the same asymptotic formula with a weaker error term $O_E(1\mathrm{/}(\log x{)}^{1\mathrm{/}14})$ is established unconditionally. These improve some recent results of Freiberg and Kurlberg.