In an introductory note of local Langlands correspondence http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/old_introductory_notes_on_local_langlands.pdf, section $11$ describes a recipe to elliptic curves:
If $E$ is a modular elliptic curve over $\mathbb Q$ coming from a modular form $f$, then one can ask how the possible local behaviours of $E$ at a prime $p$ match up with the admissible representation $V_{f,p}$ associated to $f$. Here’s how it works. Choose a prime $p$.
1) $V_{f,p}$ is unramified principal series iff $E$ has good reduction at $p$. This is because these two cases are the only cases where the conductor is $0$, so they must match up.
2) $V_{f,p}$ is special iff E has potentially multiplicative reduction at $p$. This is because these are the only cases where the image of inertia in the associated Galois representation is infinite, so they must match up.
3) $V_{f,p}$ is special associated to an unramified character iff E has multiplicative reduction. This is because these are the only two subcases of case $2)$ where the conductor is $1$.
4) $V_{f,p}$ is ramified principal series or supercuspidal iff E has bad, but potentially good, reduction. This is because these are the only cases left.
This is a easy case-by-case verification e.g we know the associated $GL_2(\mathbb Q_p)$ representation has same conductor as $E / \mathbb Q_p$, and $p||N$ iff $E$ has multiplicative reduction at $p$. More interestingly,
5) $V$ is ramified principal series iff E attains good reduction over an abelian extension of $\mathbb Q_p$.
6) $V$ is supercuspidal iff E attains good reduction over a non-abelian extension of $\mathbb Q_p$.
Why are $5)$ and $6)$ true? Do above discussions hold if $p=2,3$?
In general, given a elliptic curve over a $p$-adic field $F$, assume we know the associated irreducible $GL_2(F)$-representation of $E$, can we tell the reduction type of $E$ over all finite extensions over $F$ in an explicit way as above (and even more precisely)? I am also interested in the case $p=2,3$.
