Let $S$ be a rational surface over $\Bbb{C}$ and $\pi:S\to\Bbb{P}^1$ an elliptic fibration with Kodaira configuration $(\text{III}^*,3\text{I}_1)$.
Let $E$ be the generic fiber, which is an elliptic curve over $k(\Bbb{P}^1)$. According to Persson's classification, $E$ is isomorphic to $\Bbb{Z}$ as a group.
Considering the Mordell-Weil lattice $(E/E_\text{tor},\langle\cdot,\cdot\rangle)$, I'm trying to find the height $h(P):=\langle P,P\rangle$ of a generator $P\in E$.
It's a well-known fact that $E$ is generated by elements disjoint from the neutral section $O$, so we may assume $P\cdot O=0$, so by the height formula $h(P)=2-\text{contr}(P)$, where
$$ \text{contr}(P)=\begin{cases} 0,&\text{ if }P,O\text{ hit the same component of }\text{III}^*.\\ 3/2,&\text{ otherwise}. \end{cases} $$
So $h(P)$ is either $2$ or $1/2$. How do I know whether or not $P,O$ hit the same component?