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Let $E$ be an elliptic curve defined over ${\mathbb Q}$. For a prime $p$ ofgood 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 $C_E\in (0, 1)$ suchthat $$ \frac{1}{\pi(x)} \sum_{p\le x} e_p = 1/2 C_E x + O_E\big(x^{5/6} (\logx)^{4/3}\big) $$ for all $x\ge 2$, where the implied constant depends on $E$ atmost. When $E$ has complex multiplication, the same asymptotic formula with aweaker error term $O_E(1/(\log x)^{1/14})$ is established unconditionally.These improve some recent results of Freiberg and Kurlberg.

The average exponent of elliptic curves modulo $p$