At the manypoint page for $2^3$, genus=3, there is the note: "In his Harvard notes, Serre notes that a model of the Klein curve gives an example of a genus-3 curve with 24 points over $F_8$:
$(x + y + z)^4 + (x*y + x*z + y*z)^2 + x*y*z*(x + y + z)$
Also, $x^4 + y^4 + z^4 = 0$ has 28 points over $F_9$.
There are many available pictures for the Klein curve. Also, there are several famous graphs in graph theory associated with the Klein quartic.
Manypoints lists other curves that achieve a maximal number of points, with the lower bound and upper bound the same. Examples
$11$ genus=2 bound=24 -- $y^2=x^6+6*x^4+6*x^2+1$
$19$ genus=1 bound=28 -- $y^2=x^3+8$
$2^4$ genus=2 bound=33 -- $x^2*y^4 + x^4*y^2 + y^3 + y + 1=0$
$5^2$ genus=1 bound=36 -- $y^2=x^3+1$
$19$ genus=2 bound=36 -- $y^2=x^6+15*x^4+15*x^2+1$
$29$ genus=1 bound=40 -- $y^2=x^3+4*x$
$23$ genus=2 bound=42 -- $y^2=x^6+x^4+12*x^2+11$
$2^5$ genus=1 bound=44 -- $
y^2+x*y+y=x^3+x^2+x$
$29$ genus=2 bound=50 -- $y^2=x^6+13*x^4+13*x^2+1$
$37$ genus=1 bound=50 -- $y^2=x^3+2*x$
$43$ genus=1 bound=57 -- $y^2=x^3+9$
$7^2$ genus=1 bound=64 -- $y^2=x^3+x$
$5^2$ genus=4 bound=66 -- $y^5 + y = x^3$
$61$ genus=1 bound=77 -- $y^2=x^3+6*x+29$
$7^2$ genus=5 bound=120 -- $y^8=x^2(1-x^2)$
$11^2$ genus=1 bound=144 -- $y^2=x^3+x$
Are there good pictures of these curves over finite fields, or is there a good way of making them? Each corresponds to some number of rational points. Do these points connect up in some natural way to make a strongly regular graph, or some other famous graph? That happens with the Klein quartic. The famous graphs related to the given numbers of rational points currently don't have spectacular representations, but maybe some of these maximally symmetric surfaces align with the maximally symmetric graphs.
So, how can these curves over finite fields be made into pictures?