Properties

Label 3.12.9.1
Base \(\Q_{3}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(9\)
Galois group $D_4 \times C_3$ (as 12T14)

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Defining polynomial

\( x^{12} - 6 x^{8} + 9 x^{4} - 27 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $9$
Discriminant root field: $\Q_{3}(\sqrt{3*})$
Root number: $-i$
$|\Aut(K/\Q_{ 3 })|$: $6$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{3})$, 3.3.0.1, 3.4.3.2, 3.6.3.1

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

Unramified/totally ramified tower

Unramified subfield:3.3.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{3} - x + 1 \)
Relative Eisenstein polynomial:$ x^{4} + 3 t x^{2} + 3 t \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times D_4$ (as 12T14)
Inertia group:Intransitive group isomorphic to $C_4$
Unramified degree:$6$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:$x^{12} - 5 x^{11} + 6 x^{10} + 9 x^{9} - 26 x^{8} + 15 x^{7} - x^{6} + 12 x^{5} - 6 x^{4} - 16 x^{3} + 5 x^{2} + 6 x + 1$