scu.4.8.1.31.j
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
\( v(N) = 8\)
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
4
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 4 })^\times/U_{\mathfrak{p}^{ 4 } }\)
:
\(\begin{array}{l}
\chi^A\left((2a^{2} + 6a + 7)b - 6a + 2 \right) &= i^{ 1 }
\\
\chi^A\left((-2a^{2} + 6)b + 8a^{2} + 4a - 3 \right) &= i^{ 0 }
\\
\chi^A\left((4a^{2} + 4a + 6)b + 2a^{2} + 2a - 5 \right) &= i^{ 0 }
\\
\chi^A\left((2a^{2} + 4a - 2)b + 8a^{2} - 3 \right) &= i^{ 3 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{8} + 4a x^{7} + (134a^{2} + 804a + 584 )x^{6} + (364a^{2} + 340a + 636 )x^{5} + (297a^{2} + 43a + 480 )x^{4} + (652a^{2} + 520a + 660 )x^{3} + (378a^{2} + 706a + 476 )x^{2} + (428a^{2} + 756a + 512 )x + 997a^{2} + 85a + 24 \)