scu.3.2.1.7.a
Base Field
\(F = \mathbb{Q}_2\)
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Description
supercuspidal unramified
Construction
\(
\tau = \chi \oplus \chi^{-1}
\), with the character \(\chi\) as below
Semistability defect
\( e = 3\)
Conductor exponent
\( v(N) = 2\)
Character Order
3
Inducing Field
The inertial type \(\tau\) is reducible, induced from a character of
\(K = F(b)\), with \(b\) a root of \(x^{2} + x + 1 \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
3
Conductor exponent
1
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 1 })^\times/U_{\mathfrak{p}^{ 1 } }\)
, with \(\zeta=\frac{-1+\sqrt{-3}}{2}\) a 3rd root of unity
:
\(\begin{array}{l}
\chi^A\left(b - 5 \right) &= \zeta^{ 1 }
\\
\chi^A\left(-2b - 5 \right) &= \zeta^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{6} + 3x^{5} + 6x^{4} + 7x^{3} + 6x^{2} + 3x + 3 \)