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From what I understand, to calculate the J-invariant of an elliptic curve $E$, one can choose an involution of the elliptic curve, and quotienting out by the action of the involution defines a branched double cover of $$E\to\mathbb{CP}^1\,.$$ There are four branch points, and the j-invariant associated to the cross ratio of these 4 branch points is the j-invariant.

Is this correct so far? Then my question is: There seems to be a lot of involutions, almost one for each point of $E$ at least. Is it obvious that the j-invariant is independent of which involution you choose (or perhaps more generally independent of the branched double cover).

Also, I've read that two biholomorphic elliptic curves have branch points which can be mapped onto one another by an element of $PGL(2,\mathbb{C})\,.$ I know that $PGL(2,\mathbb{C})$ are the automorphisms of $\mathbb{CP}^1\,,$ but I don't see the relation. Does a biholomorphism of $E$ induce an automorphism on $\mathbb{CP}^1$ or something like that? Presumably I can just calculate the j-invariant for each involution and check, but I'm not sure if there isn't a better way of seeing it.

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    $\begingroup$ Hartshorne, Chapter IV, Lemma 4.4, proves that the j-invariant is independent of which involution you choose. $\endgroup$ – Kenny Wong Mar 3 '17 at 20:26

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