scu.6.4.1.15.a
Base Field
\(F = \) 3.2.1.0a1.1 \( = \mathbb{Q}_{ 3 }(a) = \mathbb{Q}_{ 3 }[x] / (x^{2} + 2 x + 2 )\)
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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 \(x^{2} + (a + 2 )x + a \)
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
:
\(\begin{array}{l}
\chi^A\left(b - 3 \right) &= \zeta^{ 1 }
\\
\chi^A\left(3b + 1 \right) &= \zeta^{ 2 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{12} + (6a + 12 )x^{11} + (12a + 30 )x^{10} + (40a + 180 )x^{9} + (210a + 534 )x^{8} + (192a + 864 )x^{7} + (1736a + 3696 )x^{6} + (6240a + 5376 )x^{5} + (10344a + 48933 )x^{4} + (43569a + 58562 )x^{3} + (7056a + 30129 )x^{2} + (11904a + 13593 )x + 58001a + 15752 \)