# Properties

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

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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: $x^{12} - 6 x^{11} + 21 x^{10} - 50 x^{9} + 93 x^{8} - 138 x^{7} + 164 x^{6} - 153 x^{5} + 111 x^{4} - 61 x^{3} + 24 x^{2} - 6 x + 1$