Galois representations and isogenies of elliptic curves Let $E$ be an elliptic curve over $\mathbb{Q}$. For each prime $\ell$, the action of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on $E[\ell]$ (the group of $\ell$-division points of $E$) defines a representation
$$\rho=\rho_\ell:\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \mathrm{GL}(2,\mathbb{F}_\ell). $$
Then how to prove that $\rho$ is reducible if and only if $E$ admits an isogeny of degree $\ell$?

The Galois representation is reducible iff there is a one-dimensional Galois-stable $\mathbb{F}_{\ell}$-subspace, say $C$.  Then $E \rightarrow E/C$ is a $\mathbb{Q}$-rational isogeny.  Conversely, if $E \rightarrow E'$ is a degree $\ell$ isogeny, its kernel $C$ is a Galois-stable subgroup of $E(\overline{\mathbb{Q}})$ of order $\ell$ so gives a one-dimensional Galois-stable $\mathbb{F}_{\ell}$-subspace of $E[\ell]$.  
