scu.6.8.1.15.a
Base Field
\(F = \) 2.1.2.3a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 )\)
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Description
supercuspidal unramified
Construction
\(
\tau = \chi \oplus \chi^{-1}
\), with the character \(\chi\) as below
Semistability defect
\( e = 6\)
Conductor exponent
\( v(N) = 8\)
Character Order
6
Inducing Field
The inertial type \(\tau\) is reducible, induced from a character of
\(K = F(b)\), with \(b\) a root of \(x^{2} + x + 1 \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
6
Conductor exponent
4
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 4 })^\times/U_{\mathfrak{p}^{ 4 } }\)
, with \(\zeta=\frac{1+\sqrt{-3}}{2}\) a 6th root of unity
:
\(\begin{array}{l}
\chi^A\left((4a - 7)b + 3a + 1 \right) &= \zeta^{ 2 }
\\
\chi^A\left(6a\cdot b + 1 \right) &= \zeta^{ 3 }
\\
\chi^A\left((4a + 6)b + 4a + 7 \right) &= \zeta^{ 3 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{12} + 6 x^{11} + 21 x^{10} + (8a + 6 )x^{9} + (4a + 4 )x^{8} + (8a + 18 )x^{7} + (2a + 19 )x^{6} + (14a + 20 )x^{5} + (11a + 8 )x^{4} + (16a + 28 )x^{3} + (25a + 17 )x^{2} + (20a + 16 )x + 26a + 31 \)