scu.6.4.1.15.a
Base Field
\(F = \) 2.1.2.3a1.4 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 3\cdot 2 )\)
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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
\( v(N) = 4\)
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
2
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 2 })^\times/U_{\mathfrak{p}^{ 2 } }\)
, with \(\zeta=\frac{1+\sqrt{-3}}{2}\) a 6th root of unity
:
\(\begin{array}{l}
\chi^A\left(-7b + 3a - 7 \right) &= \zeta^{ 5 }
\\
\chi^A\left(4b + 1 \right) &= \zeta^{ 0 }
\\
\chi^A\left(2b + 4a + 5 \right) &= \zeta^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{12} + 6 x^{11} + 21 x^{10} + (14a + 22 )x^{9} + (31a + 28 )x^{8} + (30a + 2 )x^{7} + (29a + 5 )x^{6} + (29a + 26 )x^{5} + (16a + 12 )x^{4} + (21a + 30 )x^{3} + (11a + 19 )x^{2} + (5a + 16 )x + 1 \)