If $X$ is an elliptic curve, let $P,Q\in X$, then $|P+Q|$ determines a morphism $g:X\to \mathbb{P}^1$. It is easy to see $K(X)/K(\mathbb{P}^1)$ is a Galois extension of degree 2. Let $\sigma$ be the nontrivial element in $Gal(K(X)/K(\mathbb{P}^1))$, then $\sigma$ induces an automorphism of $X$ which maps each fibre of $g$ to itself.
My question is: how do we know $\sigma$ indeed interchanges the two elements in each fibre?
My thanks in advance.