Let $\mathscr{G}=Gal(\bar{\mathbb{Q}} / \mathbb{Q})$, $E$ an elliptic curve over $\mathbb{Q}$, and consider the $\ell$-adic representation
$$ \varphi_{\ell}: \mathscr{G} \longrightarrow \mathrm{Aut}(E[\ell]) $$
I don't see why there exists an element $c \in \varphi_{\ell}(\mathscr{G})$ with eigenvalues $\{1,-1\}$. Can someone explain me this?