Suppose that $X$ is a smooth threefold, and $C \subset X$ a smooth curve. Let $Y$ be the blowup of $X$ along $C$, with exceptional divisor $E$. What is the intersection number $E^3$ on $Y$? (in terms of the genus and normal bundle of $C$, etc)

I assume that I could extract the answer from Theorem 6.7 of Fulton's book on intersection theory, were I better familiar with the contents of chapters one through five -- I'd be happy to hear either a direct method or a pointer about how to get it from Fulton!