# Properties

 Label 2.12.16.11 Base $$\Q_{2}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$16$$ Galois group $C_3\times (C_3 : C_4)$ (as 12T19)

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## Defining polynomial

 $$x^{12} + 20 x^{10} - 44 x^{8} - 4 x^{6} - 16 x^{4} - 48$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$ : $12$ Ramification exponent $e$ : $6$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $16$ Discriminant root field: $\Q_{2}(\sqrt{*})$ Root number: $i$ $|\Aut(K/\Q_{ 2 })|$: $6$ This field is not Galois over $\Q_{2}$.

## Intermediate fields

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

## Unramified/totally ramified tower

 Unramified subfield: $\Q_{2}(\sqrt{*})$ $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{2} - x + 1$$ Relative Eisenstein polynomial: $x^{6} + \left(4 t - 2\right) x^{5} - 2 x^{4} + \left(-2 t - 2\right) x^{3} + 4 t x^{2} + 4 t x + 2 t + 4 \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_3\times (C_3 : C_4)$ (as 12T19) Inertia group: Intransitive group isomorphic to $C_6$ Unramified degree: $6$ Tame degree: $3$ Wild slopes: [2] Galois mean slope: $4/3$ Galois splitting model: $$x^{12} - 39 x^{10} + 12 x^{9} + 486 x^{8} - 156 x^{7} - 2175 x^{6} - 204 x^{5} + 3738 x^{4} + 1616 x^{3} - 2031 x^{2} - 1488 x - 199$$