scu.4.2.1.31.a
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
\( e = 4\)
Conductor exponent
\( v(N) = 2\)
Character Order
4
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
4
Conductor exponent
1
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 1 })^\times/U_{\mathfrak{p}^{ 1 } }\)
:
\(\begin{array}{l}
\chi^A\left((-3a^{2} + 3a + 4)b + 3a^{2} \right) &= i^{ 1 }
\\
\chi^A\left(3b + 1 \right) &= i^{ 0 }
\\
\chi^A\left((3a - 3)b - 3a + 1 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 64 x + 19 \)