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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?

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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 (and can probably turn up a more mathematical discussion if you hunt for those Hodge numbers).

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