Properties

Label 3.12.6.2
Base \(\Q_{3}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(6\)
Galois group $C_6\times C_2$ (as 12T2)

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Defining polynomial

\( x^{12} + 108 x^{6} - 243 x^{2} + 2916 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $6$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{3}$
Root number: $1$
$|\Gal(K/\Q_{ 3 })|$: $12$
This field is Galois and abelian over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3*})$, 3.3.0.1, 3.4.2.1, 3.6.0.1, 3.6.3.1, 3.6.3.2

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

Unramified/totally ramified tower

Unramified subfield:3.6.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{6} - x + 2 \)
Relative Eisenstein polynomial:$ x^{2} + \left(-3 t^{3} - 3 t + 3\right) x - 3 t^{4} - 3 t^{3} + 3 t^{2} + 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times C_6$ (as 12T2)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$6$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$