I am interested in smooth quadric bundles $f:X \rightarrow \mathbb{P}^{1}$. Where $X$ is a smooth complex projective $3$-fold and all the fibres are smooth so that $f^{-1}(p) \cong \mathbb{CP}^{1} \times \mathbb{CP}^{1}$.
According to Beauville, for each such quadric bundle, there is a rank $4$-bundle E and a line bundle $L$ and a section $s \in H^{0}(S^{2}E^* \otimes L)$, such that $X = \{s=0\} \subset \mathbb{P}(E)$. By Grothendieck, and since tensoring by a line bundle doesn't change the projectivisation, without loss of generality we may write $E = \mathcal{O}(n_{1}) \oplus \mathcal{O}(n_{2}) \oplus \mathcal{O}(n_{3}) \oplus \mathcal{O}$ and $L = \mathcal{O}(m)$.
Question: Is there a way to compute the automorphism group of $X$ from the data $\{ n_{i},m,s \}$? b. Is there a convenient criteria to know when $X$ is toric?
Edit: Is it possible to say anything about group the subgroup $Aut_{\mathbb{P}^{1}}(X) \subset Aut(X)$ preserving fibers?