# Properties

 Label 2.12.18.31 Base $$\Q_{2}$$ Degree $$12$$ e $$4$$ f $$3$$ c $$18$$ Galois group $A_4\times C_2$ (as 12T6)

# Related objects

## Defining polynomial

 $$x^{12} + 14 x^{11} + 4 x^{10} + 12 x^{9} + 8 x^{8} + 4 x^{7} + 12 x^{6} - 8 x^{5} + 4 x^{4} + 8 x^{3} + 16 x^{2} + 16 x - 8$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $12$ Ramification exponent $e$: $4$ Residue field degree $f$: $3$ Discriminant exponent $c$: $18$ Discriminant root field: $\Q_{2}$ Root number: $1$ $|\Aut(K/\Q_{ 2 })|$: $4$ This field is not Galois over $\Q_{2}.$

## Intermediate fields

 2.3.0.1, 2.6.6.1, 2.6.6.4 x2

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

## Unramified/totally ramified tower

 Unramified subfield: 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{3} - x + 1$$ Relative Eisenstein polynomial: $$x^{4} + \left(4 t^{2} - 2 t - 2\right) x^{3} + \left(4 t^{2} + 2 t + 4\right) x^{2} + 4 x - 2 t^{2} - 2 t$$$\ \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_2\times A_4$ (as 12T6) Inertia group: Intransitive group isomorphic to $C_2^3$ Unramified degree: $3$ Tame degree: $1$ Wild slopes: [2, 2, 2] Galois mean slope: $7/4$ Galois splitting model: $x^{12} - 6 x^{11} + 12 x^{10} + 4 x^{9} - 82 x^{8} + 240 x^{7} - 424 x^{6} + 528 x^{5} - 480 x^{4} + 320 x^{3} - 152 x^{2} + 48 x - 8$