Involution of a degree-two map from a genus one curve ($\iota (p)=s-p$)

Over an algebraically closed field of characteristic different from two, a degree-two map from a smooth genus one curve to the projective line has a deck involution. After choosing an origin, all its fiber divisors on an algebraic curve are linearly equivalent and have the same group sum $s$, by the Abel-Jacobi map of a genus-one curve. The involution is consequently $p\mapsto s-p$, including its fixed ramification points. Composing two such involutions gives a translation on an elliptic curve.