# Properties

 Label 2.12.24.36 Base $$\Q_{2}$$ Degree $$12$$ e $$4$$ f $$3$$ c $$24$$ Galois group $C_2^5.(C_2\times C_6)$ (as 12T134)

# Related objects

## Defining polynomial

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

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $12$ Ramification exponent $e$: $4$ Residue field degree $f$: $3$ Discriminant exponent $c$: $24$ 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: 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{3} - x + 1$$ Relative Eisenstein polynomial: $x^{4} + \left(8 t^{2} + 4\right) x^{3} + \left(2 t^{2} - 4 t + 2\right) x^{2} + \left(8 t^{2} + 4 t\right) x + 6 t^{2} + 2 t + 6 \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_2^5.(C_2\times C_6)$ (as 12T134) Inertia group: Intransitive group isomorphic to $C_2\times C_2^2\wr C_2$ Unramified degree: $6$ Tame degree: $1$ Wild slopes: [2, 2, 2, 2, 3, 3] Galois mean slope: $87/32$ Galois splitting model: $x^{12} - 6 x^{10} + 3 x^{8} + 24 x^{6} - 36 x^{4} + 18 x^{2} - 3$