I'm trying to show that the curve $f(z,w)=z^4-w^2+1$ has genus 1. The curve is clearly non-singular, so I tried using the degree formula \begin{equation*} g=\frac{(d-1)(d-2)}{2}=\frac{3\cdot2}{2}=3, \end{equation*} but it should be equal to one. I'm not sure where my error is.
Thanks!
There are a couple of ways of doing this. First, review what the genus of a curve is. This and this question+answers might help. In particular, in the second link, look at the formula of how to adjust the formula $g=(d-1)(d-2)/2$ when there are singularities.
In this particular case, however, I would suggest you find a birational map from $C: w^2=z^4+1$ to a curve of (clearly) genus $1$. For this, look at Theorem 6, page 17, in this paper of mine.