This number of point is reached by the curve

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

where

t1:=X+Y+Z;t2:=X*Y+Y*Z+Z*X;t3:=X*Y*Z;
a1:=a^320234;a2:=a^100626;

and

a is a root of u^5+4*u+11=0 in F_{13^5}.

This curve has been found thanks to Mestre's preprint

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