  In 2009, W. D. Banks and I. E. Shparlinski studied the average densities of
primes $p \leq x$ for which the reductions of elliptic curves of small height
modulo $p$ satisfy certain arithmetic properties, namely cyclicity and
divisibility of the number of points by a fixed integer $m$. In this paper, we
refine their results, restricting the primes $p$ under consideration to lie in
an arithmetic progression $k \bmod{n}$. Furthermore, for a fixed modulus $n$,
we investigate statistical biases among the different congruence classes $k
\bmod{n}$ of primes satisfying the aforementioned properties.
