Let $E/\mathbb{C}=\mathbb{C}/\Lambda$ for a lattice $\Lambda = \mathbb{Z} + \mathbb{Z}\sqrt{5}i$. Let $\alpha=10+3\sqrt{5}i$. Show that $\alpha \in$ End$(E)$ and compute the trace and degree of $\alpha$.
We can identify the dual isogeny of $\alpha$ with its complex conjugate $\bar{\alpha}$ but I'm not sure how to use that fact to get the trace and degree of $\alpha$. What's a good way to show this inclusion and compute the trace and degree of $\alpha$?