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There are many other proofs of finding all the possible automorphism groups of elliptic curves, but I am interested in the $Hint$ and the corresponding proof in the following exercise from Vakil's book.

Ex 19.10.E. Suppose $(E,e)$ is an elliptic curve over an algebraically closed field $k$ of characteristic not 2. Show that the automorphism group of $(E,e)$ is isomorphic to $\mathbb{Z}/2,\mathbb{Z}/4$, or $\mathbb{Z}/6$.

Hint: reduce to the question of automorphisms of $\mathbb{P}^1$ fixing the point $\infty$ and a set of distinct three points $\{p_1,p_2,p_3\} \in \mathbb{P}^1- \{\infty\}$.

So the proof is to use the fact that when $k=\bar{k}$ and $\text{char}~k \neq 2$, the elliptic curve is a double cover of $\mathbb{P}^1$ branched over $\{\infty,p_1,p_2,p_3\}$. So my question is how to show the 1-1 bijection between the automorphisms of $(E,e)$ and the automorphisms of $\mathbb{P}^1$ that fix $\{\infty\}$ and $\{p_1,p_2,p_3\}$, and what are they?

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See Harthstorne, page 321, Corollary 4.7 and page 318, Lemma 4.4.

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