  Let $\delta(\Pc) = (\delta_0, \delta_1,..., \delta_d)$ be the $\delta$-vector
of an integral polytope $\Pc \subset \RR^N$ of dimension $d$. Following the
previous work of characterizing the $\delta$-vectors with $\sum_{i=0}^d
\delta_i \leq 3$, the possible $\delta$-vectors with $\sum_{i=0}^d \delta_i =
4$ will be classified. And each possible $\delta$-vectors can be obtained by
simplices. We get this result by studying the problem of classifying the
possible integral simplices with a given $\delta$-vector $(\delta_0,
\delta_1,..., \delta_d)$, where $\sum_{i=0}^d \delta_i \leq 4$, by means of
Hermite normal forms of square matrices.
