Terminology

This should be enough to help you guess my notation:

• $\Phi_{E,p}$ denotes the component group of the Néron model of $E$ at $p$.
• $N_E$ is the conductor of $E$.
• A prime $p$ is rigid for $E$ if
  $$p\nmid 2\cdot N_E\cdot \prod_{\ell\mid N_E}\char93 \Phi_{E,\ell}(\overline{\mathbb{F}}_\ell)$$
  and
  $$\rho_{E,p}:G_\mathbb{Q}\rightarrow \Aut(E[p])$$
  is irreducible.