Properties

Label 3.12.12.23
Base \(\Q_{3}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(12\)
Galois group $S_3 \times C_4$ (as 12T11)

Related objects

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Defining polynomial

\( x^{12} + 21 x^{11} + 21 x^{10} + 63 x^{9} + 36 x^{8} + 54 x^{7} + 90 x^{6} + 81 x^{3} - 81 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $-1$
$|\Aut(K/\Q_{ 3 })|$: $4$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, 3.3.3.2, 3.4.0.1, 3.6.6.3

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

Unramified/totally ramified tower

Unramified subfield:3.4.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{4} - x + 2 \)
Relative Eisenstein polynomial:$ x^{3} + \left(3 t^{3} + 3 t^{2}\right) x^{2} + \left(3 t^{3} - 3 t^{2} - 3 t\right) x + 3 t^{3} - 3 t^{2} - 3 t + 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$C_4\times S_3$ (as 12T11)
Inertia group:Intransitive group isomorphic to $S_3$
Unramified degree:$4$
Tame degree:$2$
Wild slopes:[3/2]
Galois mean slope:$7/6$
Galois splitting model:$x^{12} - 5 x^{9} + 15 x^{6} - 15 x^{3} + 5$