Properties

Label 3.12.14.14
Base \(\Q_{3}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(14\)
Galois group $(C_3\times C_3):C_4$ (as 12T17)

Related objects

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Defining polynomial

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

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $14$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $-1$
$|\Aut(K/\Q_{ 3 })|$: $6$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, 3.4.2.2, 3.6.6.1 x3

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} + \left(3 t - 3\right) x^{5} - 3 t x^{4} + 3 t x^{2} - 3 t + 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3:S_3.C_2$ (as 12T17)
Inertia group:Intransitive group isomorphic to $C_3:S_3$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:[3/2, 3/2]
Galois mean slope:$25/18$
Galois splitting model:$x^{12} - 24 x^{8} - 12 x^{6} + 189 x^{4} + 324 x^{2} + 18$