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In this paper, we investigate extreme values of $\omega(E(\mathbb{F}_p))$,where $E/\mathbb{Q}$ is an elliptic curve with complex multiplication and$\omega$ is the number-of-distinct-prime-divisors function. For fixed $\gamma >1$, we prove that \[ \#\{p \leq x : \omega(E(\mathbb{F}_p)) > \gamma\log\logx\} = \frac{x}{(\log x)^{2 + \gamma\log\gamma - \gamma + o(1)}}. \] The sameresult holds for the quantity $\#\{p \leq x : \omega(E(\mathbb{F}_p)) <\gamma\log\log x\}$ when $0 < \gamma < 1$. The argument is worked out in detailfor the curve $E : y^2 = x^3 - x$, and we discuss how the method can be adaptedfor other CM elliptic curves.

Orders of reductions of elliptic curves with many and few prime factors