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

$Gal(\mathbb{Q}(C[n])/\mathbb{Q})\to GL_2(\mathbb{Z}/n\mathbb{Z})$,

$C[n]$ 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

$\mathbb{Q}(C[n]):\mathbb{Q}$,

etc. Can one extract a concrete help on finding rational points of $C$ out of this or other statements on Galois theory?