This number is reached bu the following curve

a1*t1^4+a2*t1^2*t2+2*t1*t3+t2^2=0;

where

t1:=X+Y+Z;t2:=X*Y+X*Z+Y*Z;t3:=X*Y*Z;
a1:=a^1827;a2:=a^2281;

where a is a root of u^5+4*u+3=0 in F_{5^5}.

This curve has been found by J.-F. Mestre using his preprint

"courbes de genre 3 avec S_3 comme groupe d'automorphismes".