Properties

Label 3.12.12.25
Base \(\Q_{3}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(12\)
Galois group 12T215

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Defining polynomial

\( x^{12} + 3 x^{11} + 6 x^{10} + 6 x^{9} + 9 x^{8} - 3 x^{7} + 3 x^{6} - 9 x^{5} + 9 x^{2} - 9 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{3}$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $1$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, 3.4.2.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}(\sqrt{*})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{6} + \left(-3 t + 3\right) x^{5} - 3 t x^{4} + 3 x^{3} + \left(-3 t - 3\right) x^{2} - 3 t x - 3 t + 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:12T215
Inertia group:Intransitive group isomorphic to $C_3^2:(C_3:S_3.C_2)$
Unramified degree:$4$
Tame degree:$4$
Wild slopes:[5/4, 5/4, 5/4, 5/4]
Galois mean slope:$403/324$
Galois splitting model:$x^{12} - 3 x^{11} + 9 x^{10} - 7 x^{9} - 9 x^{8} + 30 x^{7} - 75 x^{6} + 90 x^{5} - 105 x^{4} + 95 x^{3} - 60 x^{2} + 30 x - 5$