scr.8.5.1.2.a
Base Field
\(F = \) 2.1.2.2a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 2 )\)
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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 \(x^{2} + ((a + 1)a )x + (-a - 1)a \)
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
\((\mathcal{O}_K/\mathfrak p^{ 3 })^\times/U_{\mathfrak{p}^{ 3 } }\)
:
\(\begin{array}{l}
\chi^A\left(b + 1 \right) &= i^{ 1 }
\\
\chi^A\left(-3a + 1 \right) &= i^{ 2 }
\\
\chi^A\left((2a + 2)b + a - 3 \right) &= i^{ 2 }
\\
\chi^A\left(3a\cdot b + 4a + 3 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{8} + (12a + 6 )x^{7} + (23a + 30 )x^{6} + (16a + 2 )x^{5} + 24a x^{4} + (4a + 12 )x^{3} + (14a + 10 )x^{2} + (18a + 14 )x + a + 30 \)