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Let   $E/\mathbb {Q}$   be a fixed elliptic curve. For each prime   $p$   of good reduction, write   $E(\mathbb {F}_p) \cong \mathbb {Z}/d_p \mathbb {Z} \oplus \mathbb {Z}/e_p \mathbb {Z}$  , where   $d_p \mid e_p$  . Kowalski proposed investigating the average value of   $d_p$   as   $p$   runs over the rational primes. For CM curves, he showed that   $x\log \log {x}/\log {x} \ll \sum _{p \le x} d_p \ll x\sqrt {\log {x}}.$   It was shown recently by Felix and Murty that in fact   $\sum _{p \le x} d_p$   exceeds any constant multiple of   $x\log \log {x}/\log {x}$  , once   $x$   is sufficiently large. In the opposite direction, Kim has shown that the expression   $x\sqrt {\log {x}}$   in the upper bound can be replaced by   $x\log \log {x}$  . In this paper, we obtain the correct order of magnitude for the sum:   $\sum _{p \le x} d_p \asymp x$   for all large   $x$  .

The Average of the First Invariant Factor for Reductions of CM Elliptic Curves mod p