scu.6.4.1.31.e
Base Field
\(F = \) 2.3.1.0a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{3} + x + 1 )\)
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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 \(x^{2} + a x + a \)
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
:
\(\begin{array}{l}
\chi^A\left((2a^{2} + 6a + 7)b - 6a + 2 \right) &= \zeta^{ 5 }
\\
\chi^A\left((-2a^{2} + 6)b + 8a^{2} + 4a - 3 \right) &= \zeta^{ 0 }
\\
\chi^A\left((4a^{2} + 4a + 6)b + 2a^{2} + 2a - 5 \right) &= \zeta^{ 3 }
\\
\chi^A\left((2a^{2} + 4a - 2)b + 8a^{2} - 3 \right) &= \zeta^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{12} + 6a x^{11} + (15a^{2} + 6a )x^{10} + (10a^{2} + 1012a + 976 )x^{9} + (36a^{2} + 913a + 30 )x^{8} + (886a^{2} + 1018a + 898 )x^{7} + (203a^{2} + 814a + 367 )x^{6} + (598a^{2} + 138a + 484 )x^{5} + (586a^{2} + 117a + 697 )x^{4} + (440a^{2} + 752a + 904 )x^{3} + (1003a^{2} + 994a + 147 )x^{2} + (856a^{2} + 126a )x + 929a^{2} + 132a + 925 \)