  In this paper we show that the maximum number of rational points possible for
a smooth, projective, absolutely irreducible genus 4 curve over a finite field
F_7 is 24. It is known that a genus 4 curve over F_7 can have at most 25
points. In this paper we prove that such a curve can have at most 24. On the
other hand we provide an explicit example of a genus 4 curve over F_7 having 24
points.
