Suppose that $f : X \to T$ is a family of elliptic curves over a $1$-dimensional base, and that $\sigma : T \to X$ is a section. For how many $t$ in the base do we expect $\sigma(t)$ to be $n$-torsion on $X_t$? (And what properties of the family does this depend on?)
I am most interested in the case that $f$ is just a pencil of plane cubics, so that the base locus gives $9$ sections. It seems to me that if we fix one of the sections as $0$, then another of the nine sections $\sigma$ becomes 2-torsion for exactly one value of $t \in T$. (Because the line between the point $0$ and the point $\sigma$ is tangent to exactly one member of the pencil at $\sigma$.) But for how many parameter values does this section $\sigma$ become $3$-torsion?