in \[STZ\]):

*(i) If the iterated Kodaira–Spencer map $\theta^{1,1}\theta^{2,0}$ of $f$ is non-zero, then $X$ is rigid (i.e., $h^{2,1}=0$) and birational to the Nikulin–Kummer construction of a symmetric square of a family of elliptic curves $f: E\to {{\mathbb P}}^1$. After passing to a double cover $E^{\prime}\to E$ (if necessary), the family $g^{\prime}: E^{\prime}\to
{{\mathbb P}}^1$ is one of the six modular families of elliptic curves on the Beauville’s list (\[B\]).*

\(ii) If the iterated Kodaira–Spencer map $\theta^{1,1}\theta^{2,0}$ of $f$ is zero, then $X$ is a non-rigid Calabi–Yau threefold (i.e., $h^{2,1}\neq 0$), the general fibers have Picard number at least $18$, and ${{\mathbb P}}^1\setminus
S\simeq \mathfrak H/\Gamma$ where $\Gamma$ is a congruence subgroup of $PSL(2,{{\mathbb Z}})$ of index $24$.

[**Remark**]{}: The base curve ${{\mathbb P}}^1\setminus S$ is a modular variety of genus zero, i.e., $\mathfrak H/\Gamma$ where $\Gamma$ is a torsion-free genus zero congruence subgroup of $PSL(2,{{\mathbb Z}})$ of index $12$ in case (i), and of index $24$ in case (ii). In the paper of Sun, Tan and Zuo \[STZ\], the word “modularity” refers to this fact.

The third cohomology of each of the six rigid Calabi–Yau threefolds in Theorem 1 (i) arises from a weight $4$ modular form. In the 