I am in search of a concrete example [a concrete elliptic curve in Weierstrass form] of how Galois theory helps to find rational points on an elliptic curve. Chapter VI of Silverman and Tate discusses for instance the one-to-one homomorphism
$C[n]$ denoting the points on the elliptic curve $C$ whose order divides $n$. It is discussed also that the field of definition of $C[n]$ is a Galois extension
etc. Can one extract a concrete help on finding rational points of $C$ out of this or other statements on Galois theory?