Properties

Label 3.12.12.6
Base \(\Q_{3}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(12\)
Galois group $C_2.S_3^2$ (as 12T39)

Related objects

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Defining polynomial

\( x^{12} + 24 x^{11} - 3 x^{10} + 81 x^{9} - 18 x^{8} + 54 x^{7} + 108 x^{5} - 54 x^{4} - 27 x^{3} - 81 x - 81 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $-1$
$|\Aut(K/\Q_{ 3 })|$: $2$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, 3.4.0.1, 3.6.6.5

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

Unramified/totally ramified tower

Unramified subfield:3.4.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{4} - x + 2 \)
Relative Eisenstein polynomial:$ x^{3} + \left(3 t^{3} - 3 t^{2} - 3 t - 3\right) x^{2} + \left(-3 t^{2} - 3\right) x - 3 t^{2} - 3 t - 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2.S_3^2$ (as 12T39)
Inertia group:Intransitive group isomorphic to $C_3:S_3$
Unramified degree:$4$
Tame degree:$2$
Wild slopes:[3/2, 3/2]
Galois mean slope:$25/18$
Galois splitting model:$x^{12} - 5 x^{9} + 10 x^{6} - 10 x^{3} + 5$