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:=6*a^4 + 9*a^3 + 5*a^2 + 7*a + 4;
a1:=3*a^4 + 9*a^3 + 14*a^2 + 15*a + 10;
a2:=15*a^4 + 16*a^3 + 11*a^2 + 16*a + 1;

where a is a root of u^5 + u + 14=0 in F_{17^5}.

This curve has been found thanks to Mestre's preprint

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