 Solution

A holomorphic group endomorphism of the complex torus lifts to multiplication by a scalar $c\in \mathbb{C}$, as in Question 1. It descends exactly when $c\Lambda \subseteq \Lambda$. Applying this condition to $1$ gives $c\in \mathbb{Z}[i]$. Conversely any Gaussian integer $c$ preserves $\mathbb{Z}[i]$, since the latter is a ring. Addition of endomorphisms is addition of scalars, and composition is multiplication of scalars. Thus, as rings,The wording “endomorphism algebra” here denotes this integral endomorphism ring of an elliptic curve. If it instead meant the rational algebra ${\mathrm{End}}^0(X)=\mathrm{End}(X){\otimes }_{\mathbb{Z}}\mathbb{Q}$, that algebra would be $\mathbb{Q}(i)$.