Lefschetz classes of simple factors of the Jacobian variety of a Fermat curve of prime degree over finite fields

We give a necessary and sufficient condition in terms of a matrix for which all Tate classes are Lefschetz for simple abelian varieties over an algebraic closure of a finite field. As an application, we prove under an assumption that all Tate classes are Lefschetz for simple factors of the Jacobian variety of a Fermat curve of prime degree.