arithmetic geometry - Galois representation attached to elliptic curves

Since your representation $overline{rho}$ is defined over $mathbb F_p$, you can't do things like the Hasse bounds, since
the traces $a_{ell}$ of Frobenius elements at unramified primes are just integers mod $p$,
and so don't have a well-defined absolute value.

One thing you can do is check the determinant; this should be the mod $p$ cyclotomic character if $overline{rho}$ is to come from an elliptic curve. In general (or more precisely, if $p$ is at least 7), that condition is not sufficient (although it is sufficient if $p = 2,3$ or 5);
see the various results discussed in this paper of Frank Calegari,
for example. In particular, the proof of Theorem 3.3 in that paper should
give you a feel for what can happen in the mod $p$ Galois representation attached to
weight 2 modular forms that are not defined over $mathbb Q$, while the proof of Theorem 3.4
should give you a sense of the ramification constraints on a mod $p$ representation imposed
by coming from an elliptic curve.