  We study the number of elliptic curves, up to isomorphism, over a fixed
quartic field $K$ having a prescribed torsion group $T$ as a subgroup. Let
$T=\Z/m\Z \oplus \Z/n\Z$, where $m|n$, be a torsion group such that the modular
curve $X_1(m,n)$ is an elliptic curve. Let $K$ be a number field such that
there is a positive and finite number of elliptic curves $E_T$ over $K$ having
$T$ as a subgroup. We call such pairs $(E_T, K)$ \emph{exceptional}. It is
known that there are only finitely many exceptional pairs when $K$ varies
through all quadratic or cubic fields. We prove that when $K$ varies through
all quartic fields, there exist infinitely many exceptional pairs when
$T=\Z/14\Z$ or $\Z/15\Z$ and finitely many otherwise.
