Properties

Label 3.12.14.11
Base \(\Q_{3}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(14\)
Galois group $D_6$ (as 12T3)

Related objects

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Defining polynomial

\( x^{12} + 6 x^{11} + 21 x^{10} + 36 x^{9} + 30 x^{8} + 36 x^{7} + 3 x^{6} + 36 x^{5} + 27 x^{4} - 9 x^{2} + 36 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $14$
Discriminant root field: $\Q_{3}$
Root number: $1$
$|\Gal(K/\Q_{ 3 })|$: $12$
This field is Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3*})$, 3.3.3.1 x3, 3.4.2.1, 3.6.6.4 x3, 3.6.7.1, 3.6.7.5 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} + 3 x^{5} + \left(-3 t + 3\right) x^{4} - 3 x^{2} - 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$D_6$ (as 12T3)
Inertia group:Intransitive group isomorphic to $S_3$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:[3/2]
Galois mean slope:$7/6$
Galois splitting model:$x^{12} - 6 x^{11} + 21 x^{10} - 50 x^{9} + 93 x^{8} - 138 x^{7} + 164 x^{6} - 153 x^{5} + 111 x^{4} - 61 x^{3} + 24 x^{2} - 6 x + 1$