  In this paper we generalize minimal $p$-divisible groups defined by Oort to
$F$-crystal over an algebraically closed field of positive characteristic. We
prove a structural theorem and give an explicit formula of the Frobenius
endomorphism of the isosimple minimal $F$-crystals that are the building blocks
of minimal $F$-crystals. We then define an invariant called the minimal height
for $F$-crystals using minimal $F$-crystals and give an upper bound of the
isomorphism numbers of isosimple $F$-crystals in terms of their ranks, Hodge
slopes and Newton slopes.
