I How to confirm the function has genus 116?

[aheight]

I believe the function $w(z)$ defined implicitly below has genus 116. Is there a means of independently confirming this? I believe the software Maple has a genus calculation but I don't have Maple. Is Maple the only other option?

Thanks.

$$
\begin{align*}
f(z,w)&=(8 z-4 z^2+5 z^7-4 z^{10}+4 z^{11}-8 z^{12})\\
&+(-7+6 z-5 z^2+8 z^5-9 z^7+2 z^{12})w\\
&+(-5 z^4-3 z^{10})w^2\\
&+(8-9 z+3 z^3+3 z^5+8 z^8+4 z^{12})w^3\\
&+(7-8 z^3+6 z^7+4 z^{10}-4 z^{11})w^4\\
&+(-4 z+6 z^4+6 z^5+9 z^6+6 z^9+9 z^{11}-7 z^{12})w^5\\
&+(-2+8 z-3 z^3-5 z^4+z^6+9 z^7+5 z^{11})w^6\\
&+(3 z^3+9 z^5+5 z^6+9 z^8)w^7\\
&+(8+2 z^2+3 z^7+8 z^8-2 z^{11}+7 z^{12})w^8\\
&+(z-2 z^3-2 z^4+7 z^5-6 z^9)w^9\\
&+(9 z-9 z^2+z^4+z^6+9 z^8-7 z^9-7 z^{11})w^{10}\\
&+(-4-3 z^2+3 z^3+z^5+3 z^6+7 z^8)w^{11}\\
&+(-8 z+z^4)w^{12}=0
\end{align*}
$$

 

[member 587159]

Hi. Can you clarify what "genus 116" means?

 

[aheight]

Hi. Can you clarify what "genus 116" means?

The normal Riemann surface for $w(z)$ is a donut with 116 holes.

 

[aheight]

Here is Magma code to compute the genus:

> k := Rationals();
> A<z,w> := AffineSpace(k,2);
> A;
> C := Curve(A,(8* z-4* z^2+5* z^7-4* z^10+4* z^11-8* z^12)
+(-7+6* z-5* z^2+8* z^5-9* z^7+2* z^12)*w
+(-5* z^4-3* z^10)*w^2+(8-9* z+3* z^3+3* z^5+8* z^8+4* z^12)*w^3+
(7-8* z^3+6* z^7+4* z^10-4* z^11)*w^4
+(-4* z+6* z^4+6* z^5+9* z^6+6* z^9+9* z^11-7* z^12)*w^5
+(-2+8* z-3* z^3-5* z^4+z^6+9* z^7+5* z^11)*w^6
+(3* z^3+9* z^5+5* z^6+9* z^8)*w^7
+(8+2* z^2+3* z^7+8* z^8-2* z^11+7* z^12)*w^8
+(z-2* z^3-2* z^4+7* z^5-6* z^9)*w^9
+(9* z-9* z^2+z^4+z^6+9* z^8-7* z^9-7* z^11)*w^10
+(-4-3* z^2+3* z^3+z^5+3* z^6+7* z^8)*w^11
+(-8* z+z^4)*w^12);
> C;
>Genus(C);

Magma takes 60 seconds and reports the following:

Affine Space of dimension 2 over Rational Field
Variables: z, w
Curve over Rational Field defined by
7*z^12*w^8 - 7*z^12*w^5 + 4*z^12*w^3 + 2*z^12*w - 8*z^12 - 7*z^11*w^10 -
    2*z^11*w^8 + 5*z^11*w^6 + 9*z^11*w^5 - 4*z^11*w^4 + 4*z^11 + 4*z^10*w^4 -
    3*z^10*w^2 - 4*z^10 - 7*z^9*w^10 - 6*z^9*w^9 + 6*z^9*w^5 + 7*z^8*w^11 +
    9*z^8*w^10 + 8*z^8*w^8 + 9*z^8*w^7 + 8*z^8*w^3 + 3*z^7*w^8 + 9*z^7*w^6 +
    6*z^7*w^4 - 9*z^7*w + 5*z^7 + 3*z^6*w^11 + z^6*w^10 + 5*z^6*w^7 + z^6*w^6 +
    9*z^6*w^5 + z^5*w^11 + 7*z^5*w^9 + 9*z^5*w^7 + 6*z^5*w^5 + 3*z^5*w^3 +
    8*z^5*w + z^4*w^12 + z^4*w^10 - 2*z^4*w^9 - 5*z^4*w^6 + 6*z^4*w^5 -
    5*z^4*w^2 + 3*z^3*w^11 - 2*z^3*w^9 + 3*z^3*w^7 - 3*z^3*w^6 - 8*z^3*w^4 +
    3*z^3*w^3 - 3*z^2*w^11 - 9*z^2*w^10 + 2*z^2*w^8 - 5*z^2*w - 4*z^2 - 8*z*w^12
    + 9*z*w^10 + z*w^9 + 8*z*w^6 - 4*z*w^5 - 9*z*w^3 + 6*z*w + 8*z - 4*w^11 +
    8*w^8 - 2*w^6 + 7*w^4 + 8*w^3 - 7*w
116