scu.6.6.1.31.d
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
\( v(N) = 6\)
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
3
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 3 })^\times/U_{\mathfrak{p}^{ 3 } }\)
, with \(\zeta=\frac{1+\sqrt{-3}}{2}\) a 6th root of unity
:
\(\begin{array}{l}
\chi^A\left((2a^{2} + 6a + 7)b - 6a + 2 \right) &= \zeta^{ 2 }
\\
\chi^A\left((-2a^{2} + 6)b + 8a^{2} + 4a - 3 \right) &= \zeta^{ 3 }
\\
\chi^A\left((4a^{2} + 4a + 6)b + 2a^{2} + 2a - 5 \right) &= \zeta^{ 3 }
\\
\chi^A\left((2a^{2} + 4a - 2)b + 8a^{2} - 3 \right) &= \zeta^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{12} + 6a x^{11} + (15a^{2} + 6a )x^{10} + (30a^{2} + 1004a + 1004 )x^{9} + (949a + 964 )x^{8} + (958a^{2} + 910a + 970 )x^{7} + (905a^{2} + 946a + 7 )x^{6} + (910a^{2} + 66 )x^{5} + (934a^{2} + 135a + 75 )x^{4} + (110a^{2} + 110a + 90 )x^{3} + (1003a^{2} + 970a + 925 )x^{2} + (982a^{2} + 54 )x + 5a^{2} + 24a + 11 \)