# Properties

 Label 2.12.16.17 Base $$\Q_{2}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$16$$ Galois group 12T208

# Related objects

## Defining polynomial

 $$x^{12} + 9 x^{8} - 224 x^{6} + 187 x^{4} - 32 x^{2} - 133$$

## 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: $-1$ $|\Aut(K/\Q_{ 2 })|$: $2$ 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: $x^{6} + \left(2 t + 2\right) x^{5} + \left(2 t + 2\right) x^{3} + 2 t + 2 \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: 12T208 Inertia group: Intransitive group isomorphic to $C_2^2\times C_2^4:C_3$ Unramified degree: $6$ Tame degree: $3$ Wild slopes: [4/3, 4/3, 4/3, 4/3, 2, 2] Galois mean slope: $175/96$ Global splitting model: $$x^{12} + 261 x^{8} + 1620 x^{6} + 2727 x^{4} - 180 x^{2} + 3$$