scu.6.6.1.31.a
Base Field
\(F = \) 3.1.3.4a1.1 \( = \mathbb{Q}_{ 3 }(a) = \mathbb{Q}_{ 3 }[x] / (x^{3} + 3 x^{2} + 3 )\)
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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
:
\(\begin{array}{l}
\chi^A\left((a^{2} + 4a - 2)b + 2a^{2} - 2a - 1 \right) &= \zeta^{ 3 }
\\
\chi^A\left((3a - 3)b + 4 \right) &= \zeta^{ 0 }
\\
\chi^A\left(a^{2}b + 1 \right) &= \zeta^{ 2 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{12} + 12 x^{11} + 72 x^{10} + 37 x^{9} + (5a^{2} + 51 )x^{8} + (13a^{2} + 12 )x^{7} + (19a^{2} + 7a + 32 )x^{6} + (13a^{2} + 15a + 24 )x^{5} + (9a^{2} + 15a + 78 )x^{4} + (3a^{2} + 23a + 35 )x^{3} + (6a^{2} + 12a + 6 )x^{2} + (10a^{2} + 9a + 72 )x + 5a^{2} + 3a + 22 \)