scu.4.6.1.31.w
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
\( e = 4\)
Conductor exponent
\( v(N) = 6\)
Character Order
4
Inducing Field
The inertial type \(\tau\) is reducible, induced from a character of
\(K = F(b)\), with \(b\) a root of \(x^{2} + a x + a \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
3
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 3 })^\times/U_{\mathfrak{p}^{ 3 } }\)
:
\(\begin{array}{l}
\chi^A\left((2a^{2} + 6a + 7)b - 6a + 2 \right) &= i^{ 2 }
\\
\chi^A\left((-2a^{2} + 6)b + 8a^{2} + 4a - 3 \right) &= i^{ 2 }
\\
\chi^A\left((4a^{2} + 4a + 6)b + 2a^{2} + 2a - 5 \right) &= i^{ 1 }
\\
\chi^A\left((2a^{2} + 4a - 2)b + 8a^{2} - 3 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{8} + (4a + 8 )x^{7} + (18a^{2} + 76a + 52 )x^{6} + (252a^{2} + 404a + 160 )x^{5} + (669a^{2} + 959a + 886 )x^{4} + (696a^{2} + 356a + 388 )x^{3} + (940a^{2} + 258a + 228 )x^{2} + (992a^{2} + 652a + 792 )x + 155a^{2} + 957a + 1016 \)