scu.6.4.1.31.g
Base Field
\(F = \) 3.3.1.0a1.1 \( = \mathbb{Q}_{ 3 }(a) = \mathbb{Q}_{ 3 }[x] / (x^{3} + 2 x + 1 )\)
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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 \(x^{2} + (2a + 1 )x + a \)
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
:
\(\begin{array}{l}
\chi^A\left((-3a^{2} + 3a + 4)b + 3a^{2} \right) &= \zeta^{ 5 }
\\
\chi^A\left(3b + 1 \right) &= \zeta^{ 0 }
\\
\chi^A\left((3a - 3)b - 3a + 1 \right) &= \zeta^{ 4 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{12} + (12a + 6 )x^{11} + (36a^{2} + 18a + 59040 )x^{10} + (58749a^{2} + 58879a + 59029 )x^{9} + (58356a^{2} + 3396a + 1707 )x^{8} + (6324a^{2} + 3336a + 222 )x^{7} + (43855a^{2} + 30739a + 48926 )x^{6} + (39426a^{2} + 42582a + 35364 )x^{5} + (19446a^{2} + 57438a + 54207 )x^{4} + (39391a^{2} + 1269a + 54923 )x^{3} + (39078a^{2} + 16338a + 9921 )x^{2} + (16686a^{2} + 48966a + 25026 )x + 50464a^{2} + 20509a + 46819 \)