  Given an elliptic curve $E$ over a finite field $\mathbb{F}_q$ we study the
finite extensions $\mathbb{F}_{q^n}$ of $\mathbb{F}_q$ such that the number of
$\mathbb{F}_{q^n}$-rational points on $E$ attains the Hasse upper bound. We
obtain an upper bound on the degree $n$ for $E$ ordinary using an estimate for
linear forms in logarithms, which allows us to compute the pairs of isogeny
classes of such curves and degree $n$ for small $q$. Using a consequence of
Schmidt's Subspace Theorem, we improve the upper bound to $n\leq 11$ for
sufficiently large $q$. We also show that there are infinitely many isogeny
classes of ordinary elliptic curves with $n=3$.
