TITLE: Elliptic curves and supercuspidal representations of conductor $p^2$
QUESTION [13 upvotes]: Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $p \geq 5$ be a prime of additive reduction for $E$.
Let $f$ be the newform associated to $E$, and let $\pi$ be the irreducible admissible representation of $G=\mathrm{GL}_2(\mathbf{Q}_p)$ associated to $f$ (the so-called local component of $f$ at $p$). Then $\pi$ has conductor $p^2$. Assume that $\pi$ is supercuspidal. By the classification of supercuspidal representations, there exists an irreducible representation $\xi : \mathrm{GL}_2(\mathbf{F}_p) \to \mathrm{GL}(V)$ such that $\pi \cong \mathrm{Ind}_K^G \xi$ where $K$ is the maximal compact-mod-center subgroup of $G$ given by $K=\mathbf{Q}_p^\times \cdot \mathrm{GL}_2(\mathbf{Z}_p)$. The representation $\xi$ has dimension $p-1$.
The classification of irreducible representations of $\mathrm{GL}_2(\mathbf{F}_p)$ is well-known, and in our case $\xi$ arises from a character $\phi : \mathbf{F}_{p^2}^\times \to \mathbf{C}^\times$. More precisely we have the relation $\operatorname{Tr}(\xi(g)) = - (\phi(g)+\phi(g^p))$ for every element $g$ in $\mathbf{F}_{p^2}$ not in $\mathbf{F}_p$.
Now since $f$ arises from an elliptic curve, the representation $\xi$ has trivial central character so that $\phi |_{\mathbf{F}_p^\times}=1$. This implies $\phi$ has order dividing $p+1$ and $\operatorname{Tr}(\xi(g)) = - (\phi(g)+\overline{\phi}(g))$. Moreover the representation $\xi$ can be realized over $\mathbf{Q}$, which implies that $\phi+\overline{\phi}$ takes values in $\mathbf{Q}$. Therefore the possibilities are very restricted: $\phi$ has order 3, 4 or 6. Note that this implies $p \equiv -1 \mod{} 3, 4 \textrm{ or } 6$ respectively.
Now comes my question: is there a simple way to tell whether $\phi$ has order 3, 4 or 6 in terms of $E$? By the local Langlands correspondence, I would expect a condition depending only on the local Galois representation associated to $E$. Moreover, when $p \equiv -1 \mod{12}$, does every possible order for $\phi$ occur?

REPLY [3 votes]: Jared Weinstein and I worked out an algorithm which explicitly computes the character $\phi: \mathbf{F}_{p^2}^\times \to \mathbf{C}^\times$ (more precisely, a conjugate pair of characters) using modular symbols. You can read about it in our paper. It's implemented in both Sage and Magma.
Here's the computation in Sage for the elliptic curve with Cremona label 121a:
sage: f = Newform('121a')
sage: Pi = LocalComponent(f, 11)
sage: Pi.species()
'Supercuspidal'
sage: Pi.characters()
[
Character of unramified extension Q_11(s)* (s^2 + 7*s + 2 = 0), of level 1, mapping s |--> d, 11 |--> 1,
Character of unramified extension Q_11(s)* (s^2 + 7*s + 2 = 0), of level 1, mapping s |--> -d + 1, 11 |--> 1
]
sage: xi1, xi2 = _
sage: xi1.base_ring()
Number Field in d with defining polynomial x^2 - x + 1
sage: xi1.multiplicative_order()
6