Selmer curve $C/ \Bbb{Q}: 3X^3+4Y^3+5Z^3=0$ is known to be a torsor of elliptic curve $E/ \Bbb{Q}: X^3+Y^3+60Z^3=0$. Here, $C$ is torsor of $E$ if only if $E$ acts simply transitively on $C$.
How $E$ acts on $C$? In other words, what is the map $E×C→C$ ?
My try : $E$ is Jacobia variety of $C$, so there is unique Abel Jacobi map $φ:C→E$. Let consider the map $E×C→C$ given by $φ^{-1}(φ(P)+Q)$, where '+' means addition of elliptic curve $E$.. This is well defined, but I cannot prove the transitivity.