scu.4.6.1.15.c
Base Field
\(F = \) 2.2.1.0a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + x + 1 )\)
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Description
supercuspidal unramified
Construction
\(
\tau = \chi \oplus \chi^{-1}
\), with the character \(\chi\) as below
Semistability defect
\( e = 4\)
Conductor exponent
\( v(N) = 6\)
Character Order
4
Inducing Field
The inertial type \(\tau\) is reducible, induced from a character of
\(K = F(b)\), with \(b\) a root of \(x^{2} + x + a \)
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
\((\mathcal{O}_K/\mathfrak p^{ 3 })^\times/U_{\mathfrak{p}^{ 3 } }\)
:
\(\begin{array}{l}
\chi^A\left((-6a + 1)b + 4a + 2 \right) &= i^{ 1 }
\\
\chi^A\left(6b - 2a + 7 \right) &= i^{ 2 }
\\
\chi^A\left((4a + 2)b - 6a - 7 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{8} + (8a + 12 )x^{7} + (52a + 30 )x^{6} + (84a + 996 )x^{5} + (1004a + 865 )x^{4} + (852a + 808 )x^{3} + (964a + 16 )x^{2} + (404a + 144 )x + 109a + 940 \)