Zeuthen's Theorem

If there is a $(\nu,\nu')$ correspondence between two curves of Genus $p$ and $p'$ and the number of Branch Points properly counted are $\beta$ and $\beta'$, then

$$\beta+2\nu'(p-1)=\beta'+2\nu(p'-1).$$

See also Chasles-Cayley-Brill Formula

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 246, 1959.