Let $E\subset \mathbb P^2$ be an elliptic curve given by $y^2=x(x-1)(x-\lambda)$. For a specific automorphism $\sigma\in Aut(E)$, how can we know is it given by some element in $PGL(2)$? For example, for any two points $P,Q$ on $E$ the associated $\sigma_{PQ}$ denotes the involution which switches the two sheets of the branch cover given by linear system $|P+Q|$ (generically any automorphism is given by composition of two of these). Can we say something about if it is linear in $\mathbb P^2$?
I guess for general $P,Q$ the $\sigma_{PQ}$ is not linear, but I don't know how to prove this.