scu.6.4.1.15.c
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
:
\(\begin{array}{l}
\chi^A\left(b - 3 \right) &= \zeta^{ 5 }
\\
\chi^A\left(3b + 1 \right) &= \zeta^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{12} + (6a + 12 )x^{11} + (59037a + 30 )x^{10} + (40a + 420 )x^{9} + (59043a + 822 )x^{8} + (57513a + 2016 )x^{7} + (5864a + 18864 )x^{6} + (31584a + 35328 )x^{5} + (36456a + 39909 )x^{4} + (22065a + 44651 )x^{3} + (7344a + 10545 )x^{2} + (30720a + 50178 )x + 2504a + 776 \)