  If $E$ is an elliptic curve defined over $\mathbb Q$ and $p$ is a prime of
good reduction for $E$, let $E(\mathbb F_p)$ denote the set of points on the
reduced curve modulo $p$. Define an arithmetic function $M_E(N)$ by setting
$M_E(N):= \#\{p: \#E(\mathbb F_p)= N\}$. Recently, David and the third author
studied the average of $M_E(N)$ over certain "boxes" of elliptic curves $E$.
Assuming a plausible conjecture about primes in short intervals, they showed
the following: for odd $N$, the average of $M_E(N)$ over a box with
sufficiently large sides is $\sim \frac{K^{\ast}(N)}{\log{N}}$ for an
explicitly-given function $K^{\ast}(N)$.
  The function $K^{\ast}(N)$ is somewhat peculiar: defined as a product over
the primes dividing $N$, it resembles a multiplicative function at first
glance. But further inspection reveals that it is not, and so one cannot
directly investigate its properties by the usual tools of multiplicative number
theory. In this paper, we overcome these difficulties and prove a number of
statistical results about $K^{\ast}(N)$. For example, we determine the mean
value of $K^{\ast}(N)$ over all $N$, odd $N$ and prime $N$, and we show that
$K^{\ast}(N)$ has a distribution function. We also explain how our results
relate to existing theorems and conjectures on the multiplicative properties of
$\# E(\mathbb F_p)$, such as Koblitz's conjecture.
