Let $\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}_p)$ be a continuous, irreducible Galois representation. Consider the set $\mathcal{L}_\rho$ of all elliptic curves $E/\mathbb{Q}$ such that the Galois representation $\rho_{E,p}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}_p)$ on the $p$-adic Tate module of $E$ is isomorphic to $\rho$.

Under what conditions on $\rho$ is the set $\mathcal{L}_\rho$ finite? Can one give an effective upper bound on the cardinality of $\mathcal{L}_\rho$ in terms of the invariants of $\rho$, such as its conductor and the weight of the associated modular form (if it exists)?