Properties

Label 3.12.6.1
Base \(\Q_{3}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(6\)
Galois group $C_{12}$ (as 12T1)

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Defining polynomial

\( x^{12} - 243 x^{2} + 1458 \)

Invariants

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

Intermediate fields

$\Q_{3}(\sqrt{*})$, 3.3.0.1, 3.4.2.2, 3.6.0.1

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

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
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:$x^{12} - x^{11} - 22 x^{10} + 14 x^{9} + 153 x^{8} - 62 x^{7} - 396 x^{6} + 84 x^{5} + 361 x^{4} - 87 x^{3} - 112 x^{2} + 37 x + 1$