scu.4.6.1.31.x
Base Field
\(F = \) 2.3.1.0a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{3} + x + 1 )\)
View on LMFDB ↗
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
\((\mathcal{O}_K/\mathfrak p^{ 3 })^\times/U_{\mathfrak{p}^{ 3 } }\)
:
\(\begin{array}{l}
\chi^A\left((2a^{2} + 6a + 7)b - 6a + 2 \right) &= i^{ 2 }
\\
\chi^A\left((-2a^{2} + 6)b + 8a^{2} + 4a - 3 \right) &= i^{ 2 }
\\
\chi^A\left((4a^{2} + 4a + 6)b + 2a^{2} + 2a - 5 \right) &= i^{ 1 }
\\
\chi^A\left((2a^{2} + 4a - 2)b + 8a^{2} - 3 \right) &= i^{ 2 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{8} + (8a^{2} + 12a + 8 )x^{7} + (110a^{2} + 32a + 1000 )x^{6} + (536a^{2} + 404a + 356 )x^{5} + (761a^{2} + 779a + 18 )x^{4} + (584a^{2} + 520a + 756 )x^{3} + (916a^{2} + 142a + 368 )x^{2} + (900a^{2} + 8a + 860 )x + 251a^{2} + 717a + 184 \)