Properties

Label 3.12.12.28
Base \(\Q_{3}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(12\)
Galois group $S_3\wr C_2$ (as 12T34)

Related objects

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Defining polynomial

\( x^{12} + 12 x^{11} - 3 x^{10} + 3 x^{9} + 3 x^{8} + 6 x^{7} + 12 x^{6} + 9 x^{5} + 9 x^{4} + 9 x + 9 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{3}$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $2$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3*})$, 3.4.2.1, 3.6.6.7

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}(\sqrt{*})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{6} - 3 t x^{5} + \left(-3 t - 3\right) x^{4} - 3 x^{3} + \left(3 t + 3\right) x^{2} - 3 x - 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$S_3\wr C_2$ (as 12T34)
Inertia group:Intransitive group isomorphic to $C_3:S_3.C_2$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:[5/4, 5/4]
Galois mean slope:$43/36$
Galois splitting model:$x^{12} - 6 x^{8} - 16 x^{6} + 57 x^{4} - 48 x^{2} + 16$