= 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. \b[Thus, as rings,]
$$
\boxed{\operatorname{End}(X)\cong\mathbb Z[i].}
$$
The wording “endomorphism algebra” here denotes this integral <endomorphism ring of an elliptic curve>. If it instead meant the rational algebra $\operatorname{End}^0(X)=\operatorname{End}(X)\otimes_{\mathbb Z}\mathbb Q$, that algebra would be $\mathbb Q(i)$.
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