# What is the Hurwitz number of an elliptic curve

One can associate a Hurwitz number to any rational function $f:X\to \mathbf{P}^1$ on a compact connected Riemann surface $X$ which ramifies over precisely FOUR points.

Suppose that $X$ is an elliptic curve and that $f$ is the "usual" rational function of degree $2$; the "usual" rational function of degree $2$ is given by the projection onto the $x$ coordinate when $X$ is given by $y^2= x^3+ax+b$. What is the Hurwitz number? Does it only depend on the $j$-invariant?

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What's the "usual" rational function of degree 2? I can only think of the $x$-coordinate map on $y^2 = x^3 + ax + b$, but that is ramified over four points: The roots of the cubic $x^3+ax+b$, and infinity. –  Ted Feb 19 '12 at 21:28
three was supposed to be four. Sorry! –  seporhau Feb 19 '12 at 21:51
Are you using the same definition of Hurwitz number as here? Definition 4.2 of math.ucdavis.edu/~osserman/rfg/290W/branched-covers.pdf That's the only definition I could find... If so, it seems like pretty straightforward combinatorics based on the results in section 3. –  Ted Feb 19 '12 at 23:20
In fact it follows from those results that there is only 1 branched cover having the same type as $f$ so I wonder if that's the question you're asking... –  Ted Feb 19 '12 at 23:26