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Let $\Lambda$ be a lattice in $\mathbb{C}$, and $E=\mathbb{C}/\Lambda$ the corresponding complex elliptic curve. Let $\bar{\Lambda}$ be the "conjugate" lattice, i.e. the one obtained by conjugating (as complex numbers) the points of $\Lambda$.

Can anything 'interesting' be said about the relationship between $E$ and $E':=\mathbb{C}/\bar{\Lambda}$?

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up vote 3 down vote accepted

Yes. The $j$-invariant of $E'$ is the complex conjugate of the $j$-invariant of $E$. In particular, $j(E) \in \mathbb{R} \iff$ $\overline{\Lambda}$ is homothetic to $\Lambda$.

You can use this fact to see that for every imaginary quadratic order $\mathcal{O}$, there is an $\mathcal{O}$-CM elliptic curve $E$ with real $j$-invariant: take $E = \mathbb{C}/\mathcal{O}$. In other terms, this shows that for any discriminant $D$, the modular polynomial $P_D(t) \in \mathbb{Z}[t]$ has at least one real root.

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