This number of point is reached by the curve

a0*s1^4+a1*s1^2*s2+a2*s4=0 where

s1:=X+Y=Z;s2:=X^2+Y^2+Z^2;s4:=X^4+Y^4+Z^4;
a0:=a^916;a1:=a^53;a2:=a^1093;

where a is a root of u^3+2*u+11=0 in F_{13^3}.

This curve has been found thanks to Mestre's preprint

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