I'm wondering about the notion of "oddness" in the theory of Galois representations. Usually, a Galois representation $$\rho : G_\mathbb{Q}\rightarrow GL_2(\mathbb{Q}_\ell)$$ is said to be odd if the image of complex conjugation $c\in G_\mathbb{Q}$ has determinant $-1$.

In particular, I'm interested in the case where $\rho = \rho_{E,\ell}$ is the Galois representation on the Tate module $T_\ell(E)$ of an elliptic curve $E/\mathbb{Q}$.

In this case, this fact is "proven" in Diamond-Shurman's A First Course in Modular Forms, but their proof goes by way of proving the statement for Galois representations associated to modular forms, and then using modularity to get the result for Galois reps on elliptic curves over $\mathbb{Q}$.

My questions are:

1. For an elliptic curve over $\mathbb{Q}$, does $\rho_{E,\ell}(c)$ have characteristic polynomial $x^2+1$?

2. If so, does (1) hold for any elliptic curve over $\mathbb{R}$? If so, and there are many proofs of this fact, I'd be interested in summaries/references to all the different proofs.

References would be much appreciated.