We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves $E$ over a prime finite field ${𝔽}_p$ of $p$ elements, such that the discriminant $D\left(E\right)$ of the quadratic number field containing the endomorphism ring of $E$ over ${𝔽}_p$ is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).

Let $\mathbf{E}$ be an elliptic curve defined over ${\mathbf{F}}_q$, the finite field of $q$ elements. We show that for some constant $\eta \>0$ depending only on $q$, there are infinitely many positive integers $n$ such that the exponent of $\mathbf{E}\left({\mathbf{F}}_{q^n}\right)$, the group of ${\mathbf{F}}_{q^n}$-rational points on $\mathbf{E}$, is at most $q^{n}exp\left(-n^{\eta /loglogn}\right)$. This is an analogue of a result of R. Schoof on the exponent of the group $\mathbf{E}\left({\mathbf{F}}_p\right)$ of ${\mathbf{F}}_p$-rational points, when a fixed elliptic curve $\mathbf{E}$ is defined over $\mathbb{Q}$ and the prime $p$ tends to infinity.

This note gives a survey of some recent results on the stable reduction of covers of the projective line branched at three points.