scu.3.10.1.31.b
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
\( e = 3\)
Conductor exponent
\( v(N) = 10\)
Character Order
3
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
3
Conductor exponent
5
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 5 })^\times/U_{\mathfrak{p}^{ 5 } }\)
, with \(\zeta=\frac{-1+\sqrt{-3}}{2}\) a 3rd root of unity
:
\(\begin{array}{l}
\chi^A\left((a^{2} + 4a - 2)b + 2a^{2} - 2a - 1 \right) &= \zeta^{ 0 }
\\
\chi^A\left((3a - 3)b + 4 \right) &= \zeta^{ 1 }
\\
\chi^A\left(a^{2}b + 1 \right) &= \zeta^{ 1 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{6} + 21a^{2} x^{5} + (15a^{2} + 78 )x^{4} + (2a + 77 )x^{3} + (12a^{2} + 30 )x^{2} + (15a + 60 )x + 22a^{2} + 23a + 20 \)