# Properties

 Label 2.12.12.33 Base $$\Q_{2}$$ Degree $$12$$ e $$4$$ f $$3$$ c $$12$$ Galois group $C_3\times S_4$ (as 12T45)

# Related objects

## Defining polynomial

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

## Invariants

 Base field: $\Q_{2}$ Degree $d$ : $12$ Ramification exponent $e$ : $4$ Residue field degree $f$ : $3$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{2}(\sqrt{*})$ Root number: $1$ $|\Aut(K/\Q_{ 2 })|$: $3$ 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: $x^{4} + 2 x^{3} + \left(2 t^{2} + 2 t\right) x^{2} + \left(2 t^{2} + 2 t\right) x + 2 t + 2 \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_3\times S_4$ (as 12T45) Inertia group: Intransitive group isomorphic to $A_4$ Unramified degree: $6$ Tame degree: $3$ Wild slopes: [4/3, 4/3] Galois mean slope: $7/6$ Galois splitting model: $x^{12} - 4 x^{9} + 54 x^{8} - 54 x^{7} + 122 x^{6} - 72 x^{5} + 93 x^{4} - 48 x^{3} + 18 x^{2} + 1$