Let $X$ be a smooth variety over $\mathbb{C}$. Blowing up a subvariety $Y\subset X$ of codimension $\ge 2$, we get $\pi: X'\rightarrow X$. Assume $X'$ is smooth and $E$ is the only exceptional divisor.

Let $C\subset E$, but not contracted by $\pi$, i.e. $\pi(C)$ is still a curve. What can we say about the intersection number $C. E$?