class of a trivial fibration vanishes, it follows that $\Theta(f) = 0$.

Our examples all have $6$ singular fibers. Hence $$\frac{1}{2}\deg \Omega^1_{{{\mathbb P}}^1}(\log S) =
\frac{1}{2}  \deg{{\mathcal O}}_{{{\mathbb P}}^1}(-2+6) = 2.$$

On the other hand, since $X$ is a Calabi–Yau variety we have $$\omega_{X/{{\mathbb P}}^1} = \omega_X\otimes
(f^*\omega_{{{\mathbb P}}^1})^{-1} =
(f^*\omega_{{{\mathbb P}}^1})^{-1}.$$ Hence $$\label{deg}
 f_*\omega_{X/{{\mathbb P}}^1} =
f_*f^*(\omega_{{{\mathbb P}}^1})^{-1}=
(f_*f^*\omega_{{{\mathbb P}}^1})^{-1}=
\omega_{{{\mathbb P}}^1}^{-1},$$ whose degree is $2$ as well, concluding the proof of Theorem 7.

[**Remarks 8**]{}:

*(1) In the other cases in Table 2 the monodromy on the points of order $2$ of $A_t[2]$ is non-trivial, and the calculation gives that the monodromy of $f$ around the cusps is not unipotent. We know by Remark 3, the groups \#1, \#5 and \#8 are in the same $PSL(2,{{\mathbb R}})$-conjugacy class. However, this group theoretic property does not guarantee isomorphisms of the corresponding Calabi–Yau threefolds, since the fiber structures are not preserved. Similarly, the groups \#4, \#6 and \#9 are $PSL(2,{{\mathbb R}})$-conjugate, but geometric structures are different (as the fibers over the cusps are different). The same applies to Examples \#2 and \#7. Therefore, Examples \#4, \#7, \#8 and \#9 are not covered by our examples. Also we do not know 