When is the local representation associated to an elliptic curve a Steinberg? If $E$ is an elliptic curve over $\mathbb{Q}$, and $\pi$ is the automorphic representation of $\mathrm{GL}_2$ associated to $E$, then one can write $\pi = \otimes_v \pi_v$ with each $\pi_v$ an irreducible representation of $\mathrm{GL}_2(\mathbb{Q}_v)$.
For a non-archimedean place $v$, how do we know the isomorphism class of $\pi_v$, in terms of informations about the curve $E$?
In particular, when is $\pi_v$ a Steinberg or a twisted Steinberg? Some explanation or reference will be welcome. Thank you!
 A: The local representation $\pi_{v}$ attached to an elliptic curve is a Steinburg or a twisted Steinburg if and only if $E$ has potentially multiplicative reduction. This follows from the discussion in Section 15 of Rohrlich's paper "Elliptic curves and the Weil-Deligne group", where it is shown that the corresponding representation of the Weil-Deligne group is a twist of the 2-dimensional special representation if and only if $E$ has potentially multiplicative reduction. (Note: Since we're over $\mathbb{Q}$, we know that $E$ is modular, the local Langlands correspondence will map
a twist of the Steinburg to a twist of the special representation and vice versa.)
Another source is the paper "Euler Factors and Local Root Number of Symmetric Powers of elliptic curves" by Dummigan, Martin, and Watkins, which can be found here. Both the paper of Rohrlich and the paper of Dummigan, Martin and Watkins also discuss the case of potentially good reduction (which can yield a supercuspidal or a principal series representation).
