scr.8.18.1.2.h
Base Field
\(F = \) 2.1.3.2a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{3} + 2 )\)
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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
\( v(N) = 18\)
Character Order
4
Inducing Field
The inertial type \(\tau\) is reducible, induced from a character of
\(K = L(b)\), with \(b\) a root of \(x^{2} - a x + (-a + 1)a \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
16
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 16 })^\times/U_{\mathfrak{p}^{ 16 } }\)
:
\(\begin{array}{l}
\chi^A\left(b + 1 \right) &= i^{ 2 }
\\
\chi^A\left((a^{2} + 4a + 4)b - 2a^{2} - 2a + 1 \right) &= i^{ 2 }
\\
\chi^A\left((3a^{2} + 2)b + 2a^{2} - 3a - 1 \right) &= i^{ 0 }
\\
\chi^A\left((3a^{2} + 4)b + 3a^{2} - 3a + 1 \right) &= i^{ 1 }
\\
\chi^A\left((-2a^{2} + 4a)\cdot b + 4a^{2} + 4a - 3 \right) &= i^{ 2 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{8} + (4a^{2} + 4a )x^{7} + (6a^{2} + 4a + 2 )x^{6} + (4a^{2} + 4a + 12 )x^{5} + (6a^{2} + 3a + 6 )x^{4} + (4a^{2} + 12 )x^{3} + (6a^{2} + 8 )x^{2} + 7a^{2} + 5a + 2 \)