ps.3.4.1.1.m
Base Field
\(F = \) 3.3.1.0a1.1 \( = \mathbb{Q}_{ 3 }(a) = \mathbb{Q}_{ 3 }[x] / (x^{3} + 2 x + 1 )\)
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Description
principal series
Construction
\(
\tau = \chi \oplus \chi^{-1}
\), with the character \(\chi\) as below
Semistability defect
\( e = 3\)
Conductor exponent
\( v(N) = 4\)
Character Order
3
Underlying Character
Character \(\chi^A:\mathcal O_F^\times \to \mathbb C^\times\) with the following properties:
Order
3
Conductor exponent
2
Values on generators of \((\mathcal{O}_F/\mathfrak p^{ 2 })^\times\)
, with \(\zeta=\frac{-1+\sqrt{-3}}{2}\) a 3rd root of unity
:
\(\begin{array}{l}
\chi^A\left(3a^{2} - 2a + 3 \right) &= \zeta^{ 1 }
\\
\chi^A\left(4 \right) &= \zeta^{ 2 }
\\
\chi^A\left(3a + 1 \right) &= \zeta^{ 2 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{3} + 6a^{2} x^{2} + 3 \)