scu.6.4.1.31.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
\( e = 6\)
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
:
\(\begin{array}{l}
\chi^A\left((-3a^{2} + 3a + 4)b + 3a^{2} \right) &= \zeta^{ 3 }
\\
\chi^A\left(3b + 1 \right) &= \zeta^{ 0 }
\\
\chi^A\left((3a - 3)b - 3a + 1 \right) &= \zeta^{ 2 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{12} + (12a + 6 )x^{11} + (36a^{2} + 66a + 15 )x^{10} + (180a^{2} + 310a + 100 )x^{9} + (1611a^{2} + 1524a + 375 )x^{8} + (6324a^{2} + 168a + 57831 )x^{7} + (13654a^{2} + 53107a + 53150 )x^{6} + (27168a^{2} + 30351a + 40341 )x^{5} + (16773a^{2} + 41835a + 19341 )x^{4} + (44401a^{2} + 49092a + 30461 )x^{3} + (14121a^{2} + 30522a + 56721 )x^{2} + (49164a^{2} + 16998a + 16503 )x + 4459a^{2} + 45610a + 39148 \)