ps.6.4.1.1.a
Base Field
\(F = \) 3.1.3.3a1.1 \( = \mathbb{Q}_{ 3 }(a) = \mathbb{Q}_{ 3 }[x] / (x^{3} + 3 x + 3 )\)
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Description
principal series
Construction
\(
\tau = \chi \oplus \chi^{-1}
\), with the character \(\chi\) as below
Semistability defect
\( e = 6\)
Conductor exponent
\( v(N) = 4\)
Character Order
6
Underlying Character
Character \(\chi^A:\mathcal O_F^\times \to \mathbb C^\times\) with the following properties:
Order
6
Conductor exponent
2
Values on generators of \((\mathcal{O}_F/\mathfrak p^{ 2 })^\times\)
, with \(\zeta=\frac{1+\sqrt{-3}}{2}\) a 6th root of unity
:
\(\begin{array}{l}
\chi^A\left(-a - 1 \right) &= \zeta^{ 5 }
\\
\chi^A\left(a^{2} + 1 \right) &= \zeta^{ 0 }
\\
\chi^A\left(-3a + 1 \right) &= \zeta^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{6} + 19a x^{4} + 17a \)