= Automorphism of a one-dimensional complex torus
Every identity-preserving <biholomorphism> of $\mathbb C/\Lambda$ lifts to multiplication by a nonzero <complex number> $\zeta$ satisfying $\zeta\Lambda=\Lambda$. Relative to a $\mathbb Z$-basis of $\Lambda$, multiplication by $\zeta$ is represented by a <unimodular matrix> $A\in\operatorname{GL}_2(\mathbb Z)$. Its real <determinant> is $|\zeta|^2$, while complex multiplication preserves orientation, so $\det A=1$ and $|\zeta|=1$. The <Cayley-Hamilton theorem> gives
$$
\zeta^2-\operatorname{tr}(A)\zeta+1=0.
$$
Consequently $2\operatorname{Re}\zeta=\operatorname{tr}(A)\in\{-2,-1,0,1,2\}$, and $\zeta$ is a <root of unity> of order $1$, $2$, $3$, $4$, or $6$.
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