scr.8.8.1.10.g
Base Field
\(F = \) 2.3.1.0a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{3} + x + 1 )\)
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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
\( v(N) = 8\)
Character Order
4
Inducing Field
The inertial type \(\tau\) is reducible, induced from a character of
\(K = L(b)\), with \(b\) a root of \(x^{2} + 2 x + (-a^{2} - 2)\cdot 2 \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
6
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 6 })^\times/U_{\mathfrak{p}^{ 6 } }\)
:
\(\begin{array}{l}
\chi^A\left(a\cdot b + 1 \right) &= i^{ 1 }
\\
\chi^A\left((4a^{2} + 4a)\cdot b - 2a^{2} + 2a + 1 \right) &= i^{ 2 }
\\
\chi^A\left((a^{2} + 4a - 3)b + a^{2} + 4a + 1 \right) &= i^{ 2 }
\\
\chi^A\left((3a^{2} - a + 3)b + a^{2} + 2a - 3 \right) &= i^{ 1 }
\\
\chi^A\left((4a + 2)b - 1 \right) &= i^{ 1 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{8} + (8a^{2} + 8 )x^{7} + 4a^{2} x^{6} + (2a^{2} + 2 )x^{4} + (8a + 8 )x^{3} + (4a + 4 )x^{2} + (8a^{2} + 8a + 8 )x + 4a + 10 \)