ps.4.8.1.1.s
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
principal series
Construction
\(
\tau = \chi \oplus \chi^{-1}
\), with the character \(\chi\) as below
Semistability defect
\( e = 4\)
Conductor exponent
\( v(N) = 8\)
Character Order
4
Underlying Character
Character \(\chi^A:\mathcal O_F^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
4
Values on generators of \((\mathcal{O}_F/\mathfrak p^{ 4 })^\times\)
:
\(\begin{array}{l}
\chi^A\left(-3a^{2} + 4a + 5 \right) &= i^{ 1 }
\\
\chi^A\left(7 \right) &= i^{ 2 }
\\
\chi^A\left(2a + 1 \right) &= i^{ 1 }
\\
\chi^A\left(-4a - 3 \right) &= i^{ 1 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{4} + (8a + 8 )x^{3} + 4 x^{2} + 18 \)