Properties

Label 3.12.12.22
Base \(\Q_{3}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(12\)
Galois group $S_3\times C_3:S_3.C_2$ (as 12T119)

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Defining polynomial

\( x^{12} + 18 x^{11} + 21 x^{10} - 69 x^{9} - 81 x^{8} + 72 x^{7} - 90 x^{6} - 108 x^{5} + 54 x^{4} - 108 x^{3} - 81 \)

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: $-1$
$|\Aut(K/\Q_{ 3 })|$: $1$
This field is not Galois over $\Q_{3}$.

Intermediate fields

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

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} + 3 t x^{2} + \left(-3 t^{2} - 3\right) x - 3 t^{3} + 3 t^{2} + 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$S_3\times C_3:S_3.C_2$ (as 12T119)
Inertia group:Intransitive group isomorphic to $C_3^3:C_2$
Unramified degree:$4$
Tame degree:$2$
Wild slopes:[3/2, 3/2, 3/2]
Galois mean slope:$79/54$
Galois splitting model:$x^{12} + 12 x^{10} - 8 x^{9} + 54 x^{8} - 72 x^{7} + 88 x^{6} - 216 x^{5} - 39 x^{4} - 152 x^{3} - 180 x^{2} + 192 x + 124$