# Mirror partners of some Calabi-Yau threefolds

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?

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).