Properties

Label 3.12.13.4
Base \(\Q_{3}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(13\)
Galois group $S_3\wr C_2$ (as 12T36)

Related objects

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Defining polynomial

\( x^{12} - 3 x^{10} + 3 x^{6} - 3 x^{5} + 3 x^{4} - 3 x^{2} - 3 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $12$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $13$
Discriminant root field: $\Q_{3}(\sqrt{3*})$
Root number: $i$
$|\Aut(K/\Q_{ 3 })|$: $2$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{3})$, 3.4.3.2, 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:$\Q_{3}$
Relative Eisenstein polynomial:\( x^{12} - 3 x^{10} + 3 x^{6} - 3 x^{5} + 3 x^{4} - 3 x^{2} - 3 \)

Invariants of the Galois closure

Galois group:$S_3\wr C_2$ (as 12T36)
Inertia group:$(C_3\times C_3):C_4$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:[5/4, 5/4]
Galois mean slope:$43/36$
Galois splitting model:$x^{12} + 6 x^{10} + 12 x^{6} + 21 x^{4} + 6 x^{2} + 6$