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:= 16*a^4 + 5*a^3 + 9*a^2 + 13*a + 17;
a1:=15*a^4 + 4*a^3 + 5*a + 9;
a2:=3*a^4 + 4*a^2 + 3*a + 15;

where a is a root of u^5+5*u+17=0 in F_{19^5}.

This curve has been found thanks to Mestre's preprint

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