Properties

Label 3.12.0.1
Base \(\Q_{3}\)
Degree \(12\)
e \(1\)
f \(12\)
c \(0\)
Galois group $C_{12}$ (as 12T1)

Related objects

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Defining polynomial

\( x^{12} - x^{4} - x^{3} - x^{2} + x - 1 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $12$
Discriminant exponent $c$ : $0$
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, 3.4.0.1, 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:3.12.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{12} + 2 x^{4} - x^{3} + 2 x^{2} - 2 x + 2 \)
Relative Eisenstein polynomial:$ x - 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Trivial
Unramified degree:$12$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{12} - x^{11} + 3 x^{10} - 4 x^{9} + 9 x^{8} + 2 x^{7} + 12 x^{6} + x^{5} + 25 x^{4} - 11 x^{3} + 5 x^{2} - 2 x + 1$