It is non-rigid, and the given fibration $f:X{\rightarrow}X(\tilde{\Gamma})$ is semi-stable, with vanishing (iterated) Kodaira–Spencer mapping. We have $$\deg{f_*\omega_{X/{{\mathbb P}}^1}} = 2 =
\frac{1}{2}\,\mbox{deg}\,\Omega^1_{{{\mathbb P}}^1}
(\mbox{log}\,S),$$ In other words, $X$ reaches the (stronger) Arakelov–Yau bound.* ]{}

[**Remark**]{}: [*For the first case in Table 2 ($\tilde{\Gamma} = \Gamma(4)$) this example is indicated in \[STZ\].*]{}

[**Proof**]{}: The Examples we chose are those in which $\tilde{\Gamma}$ is a subgroup of $\Gamma(2)$. (This is because otherwise, the points of order $2$ of $X(\Gamma)$ coincide (over the cusps).) Thus the $2$-torsion points (of $E_t$ and hence of $A_t$) are distinct for [*all*]{} $t\in
X(\tilde{\Gamma})$. It follows that the locus $A[2]$ of $2$-torsion points is smooth, and hence so is the blow-up $X$. We have $H^i(X) =
H^i(Y(\tilde{\Gamma})\times E)^{<\pm 1>}$. But $\pm 1$ acts as $\pm 1$ on both the non-trivial holomorphic $1$-form $\omega_1$ of $E$ and on the non-trivial holomorphic $2$-form $\omega_2$ of $Y(\tilde{\Gamma})$. Hence $\omega_1\wedge
\omega_2$ descends to a holomorphic $3$-form $\omega_3$ on $X$. Its divisor can only be supported on the proper transform ${\mathcal{F}}$ of $A[2]$; however $\mathcal{F}$ intersects each fiber $f^{-1}(t)$ in sixteen $(-2)$-curves, which do not contribute to the canonical class, so that $\omega_3$ is indeed nowhere-vanishing. The Künneth formula gives that 