ps.2.2.1.1.a
Base Field
\(F = \) 3.1.2.1a1.2 \( = \mathbb{Q}_{ 3 }(a) = \mathbb{Q}_{ 3 }[x] / (x^{2} - 3 )\)
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Description
principal series
Construction
\(
\tau = \chi \oplus \chi^{-1}
\), with the character \(\chi\) as below
Semistability defect
\( e = 2\)
Conductor exponent
\( v(N) = 2\)
Character Order
2
Underlying Character
Character \(\chi^A:\mathcal O_F^\times \to \mathbb C^\times\) with the following properties:
Order
2
Conductor exponent
1
Values on generators of \((\mathcal{O}_F/\mathfrak p^{ 1 })^\times\)
:
\(\begin{array}{l}
\chi^A\left(-a - 1 \right) &= (-1)^{ 1 }
\\
\chi^A\left(4 \right) &= (-1)^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{2} + a \)