It is not absolutely irreducible: over the algebraic closure, it is the direct sum of $V^{Fr ^i}$ ($0\leq i \leq e-1$), where $Fr : {\mathbb F}_q\rightarrow {\mathbb F}_q$ is the map $x\mapsto x^p$.
I think it is irreducible over ${\mathbb F}_p$ (there may be some problems if $e$ is even), but for $e$ odd, this description over the algebraic closure would imply that there are no ${\mathbb F}_p$ invariant subspaces.