I don't have experience in mirror symmetry, hence I am not sure that my question is of research level. Sorry in advance.


Let $k$ be an algebraically closed field of characteristic $\neq 2, 3$. Consider the ordinary elliptic $k$-curve $E\!: y^2 = x^3 + 1$ (of $j$-invariant $0$) and its cubic power $E^3$. There are quite natural automorphisms on $E^3$, namely
$$
\begin{array}{l}
 \alpha\_0\!: (P\_0,P\_1,P\_2) \mapsto \big( P\_0, [-1]P\_1, [-1]P\_2 \big), \\
 \alpha\_1\!: (P\_0,P\_1,P\_2) \mapsto \big( [-1]P\_0, P\_1, [-1]P\_2 \big), \\
 [\omega]^{\times 3}\!: (P\_0,P\_1,P\_2) \mapsto \big( [\omega]P\_0, [\omega]P\_1, [\omega]P\_2 \big),
\end{array}
$$
where $[-1]\!: (x,y) \mapsto (x,-y)$ and $[\omega]\!: (x,y) \mapsto (\omega x,y)$ for $\omega := \sqrt[3]{1}$, $\omega \neq 1$. It turns out that the singularity resolutions of the quotients $E^3/\langle \alpha\_0, \alpha\_1 \rangle$ and $E^3/[\omega]^{\times 3}$
(with the Galois groups $(\mathbb{Z}/2)^2$ and $\mathbb{Z}/3$ respectively) are Calabi-Yau threefolds. See, e.g., the articles <https://www.sciencedirect.com/science/article/pii/S0022314X08002011> ($\S 4$) and <https://arxiv.org/abs/1306.1590> ($\S 1.3$).


Help me please. What are their mirror partners? How can one construct them explicitly?



Mirror partners of some Calabi-Yau threefolds
Your two examples are actually of very different characters. The first has Hodge numbers $h^{2,1} = 3$ and $h^{1,1} = 51$; the second is rigid. This means that in the first case you're looking for a Calabi-Yau threefold with "mirror" Hodge numbers $h^{2,1} = 51$ and $h^{1,1} = 3$, while the mirror of the second may not be a Calabi-Yau threefold at all. You'll find a discussion of the first mirror in Marianna Larosa's physics dissertation [here](https://s3.cern.ch/inspire-prod-files-7/75d71c75e61fc252357fc79e2442d5e6) (and can probably turn up a more mathematical discussion if you hunt for those Hodge numbers).


