  Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In 1976, Lang and
Trotter conjectured an asymptotic formula for the number $\pi_{E,r}(X)$ of
primes $p \leq X$ of good reduction for which the Frobenius trace at $p$
associated to $E$ is equal to a given fixed integer $r$. We investigate
elliptic curves $E$ over $\mathbb{Q}$ that have a missing Frobenius trace, i.e.
for which the counting function $\pi_{E,r}(X)$ remains bounded as $X
\rightarrow \infty$, for some $r \in \mathbb{Z}$. In particular, we classify
all elliptic curves $E$ over $\mathbb{Q}(t)$ that have a missing Frobenius
trace.
