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

The normal order of of the divisor-counting function for invariants of rank 2 Drinfeld modules