scu.6.6.1.15.c
Base Field
\(F = \) 3.1.2.1a1.2 \( = \mathbb{Q}_{ 3 }(a) = \mathbb{Q}_{ 3 }[x] / (x^{2} - 3 )\)
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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
\( v(N) = 6\)
Character Order
6
Inducing Field
The inertial type \(\tau\) is reducible, induced from a character of
\(K = F(b)\), with \(b\) a root of \(x^{2} + 2 x + 2 \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
6
Conductor exponent
3
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 3 })^\times/U_{\mathfrak{p}^{ 3 } }\)
, with \(\zeta=\frac{1+\sqrt{-3}}{2}\) a 6th root of unity
:
\(\begin{array}{l}
\chi^A\left(-2b - 2a - 4 \right) &= \zeta^{ 5 }
\\
\chi^A\left(-3a\cdot b + 1 \right) &= \zeta^{ 0 }
\\
\chi^A\left((-4a + 3)b - 1 \right) &= \zeta^{ 4 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{12} + 12 x^{11} + 72 x^{10} + 37 x^{9} + 147 x^{8} + 213 x^{7} + (115a + 29 )x^{6} + (204a + 27 )x^{5} + 78 x^{4} + (17a + 68 )x^{3} + (78a + 144 )x^{2} + (18a + 144 )x + 66a + 61 \)