  We consider the finite set of isogeny classes of $g$-dimensional abelian
varieties defined over the finite field $\mathbb{F}_q$ with endomorphism
algebra being a field. We prove that the class within this set whose varieties
have maximal number of rational points is unique, for any prime even power $q$
big enough and verifying mild conditions. We describe its Weil polynomial and
we prove that the class is ordinary and cyclic outside the primes dividing an
integer that only depends on $g$. In dimension $3$, we prove that the class is
ordinary and cyclic and give explicitly its Weil polynomial, for any prime even
power $q$.
