scr.8.5.1.2.a
Base Field
\(F = \mathbb{Q}_2\)
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Description
supercuspidal ramified
Construction
\(
\tau = \operatorname{Ind}^{I_K}_{I_F} \chi
\), with \(K\) and \(\chi\) as below
Semistability defect
\( e = 8\)
Conductor exponent
\( v(N) = 5\)
Character Order
4
Inducing Field
The inertial type \(\tau\) is reducible, induced from a character of
\(K = L(b)\), with \(b\) a root of \(x^{2} + 2 x - 2 \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
3
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 3 })^\times/U_{\mathfrak{p}^{ 3 } }\)
:
\(\begin{array}{l}
\chi^A\left(b + 1 \right) &= i^{ 1 }
\\
\chi^A\left(-2b - 3 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{8} + 2x^{6} + 4x^{4} + 4x + 6 \)