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

Weil polynomials of abelian varieties over finite fields with many rational points