  Let $E$ be an elliptic curve over $\Q$. It is well known that the ring of
endomorphisms of $E_p$, the reduction of $E$ modulo a prime $p$ of ordinary
reduction, is an order of the quadratic imaginary field $Q(\pi_p)$ generated by
the Frobenius element $\pi_p$. When the curve has complex multiplication (CM),
this is always a fixed field as the prime varies. However, when the curve has
no CM, very little is known, not only about the order, but about the fields
that might appear as algebra of endomorphisms varying the prime. The ring of
endomorphisms is obviously related with the arithmetic of $a_p^2-4p$, the
discriminant of the characteristic polynomial of the Frobenius element. In this
paper, we are interested in the function $\pi_{E,r,h}(x)$ counting the number
of primes $p$ up to $x$ such that $a_p^2-4p$ is square-free and in the
congruence class $r$ modulo $h$. We give in this paper the precise asymptotic
for $\pi_{E,r,h}(x)$ when averaging over elliptic curves defined over the
rationals, and we discuss the relation of this result with the Lang-Trotter
conjecture, and with some other problems related to the curve modulo $p$.
