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:=2;a2:=0;

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

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