Background

We are given a curve with integer coefficients. I want to make a suggestion in another question (https://mathoverflow.net/questions/19880/computationally-bounding-a-curves-genus-from-below) into a deterministic algorithm for finding the genus of a plane curve.

The suggestion is: reduce modulo a random prime and find all singular points there. If the prime was of good reduction, then these are the reductions of all of the algebraic singular points, and you can compute the genus easily from here.

Question

What is an effective bound on the largest prime of bad reduction?

What I imagine

Say $C$ is given by $\sum a_{ij} x^i y^j$, then I imagine a bound similar to: $\displaystyle\sum_{\sigma} \prod a_{i\sigma^{-1} (i)}$