scr.12.9.1.2.d
Base Field
\(F = \) 3.1.3.4a2.2 \( = \mathbb{Q}_{ 3 }(a) = \mathbb{Q}_{ 3 }[x] / (x^{3} + 2\cdot 3 x^{2} + 4\cdot 3 )\)
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Description
supercuspidal ramified
Construction
\(
\tau = \operatorname{Ind}^{I_K}_{I_F} \chi
\), with \(K\) and \(\chi\) as below
Semistability defect
\( e = 12\)
Conductor exponent
\( v(N) = 9\)
Character Order
6
Inducing Field
The inertial type \(\tau\) is reducible, induced from a character of
\(K = F(b)\), with \(b\) a root of \(x^{2} - a \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
6
Conductor exponent
8
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 8 })^\times/U_{\mathfrak{p}^{ 8 } }\)
, with \(\zeta=\frac{1+\sqrt{-3}}{2}\) a 6th root of unity
:
\(\begin{array}{l}
\chi^A\left(-b - 1 \right) &= \zeta^{ 5 }
\\
\chi^A\left(-3a\cdot b + 1 \right) &= \zeta^{ 0 }
\\
\chi^A\left(a^{2}b + 1 \right) &= \zeta^{ 0 }
\\
\chi^A\left(-3b + 1 \right) &= \zeta^{ 2 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{12} + (12a^{2} + 9a + 27 )x^{8} + (15a^{2} + 57 )x^{4} + 8a \)