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}?