  We compute the first and second moments of the divisor-counting function for
the Euler-Poincar\'{e} characteristic and the trace of Frobenius for the
reductions modulo $p$ of a rank 2 Drinfeld module with nontrivial endomorphism
ring, as the prime $p$ varies over the primes of ordinary reduction of the
Drinfeld module. From these moments we derive the normal order of the number of
prime divisors of these invariants.
