# Properties

 Label 2.12.8.1 Base $$\Q_{2}$$ Degree $$12$$ e $$3$$ f $$4$$ c $$8$$ Galois group $C_3 : C_4$ (as 12T5)

# Related objects

## Defining polynomial

 $$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$ : $12$ Ramification exponent $e$ : $3$ Residue field degree $f$ : $4$ Discriminant exponent $c$ : $8$ Discriminant root field: $\Q_{2}(\sqrt{*})$ Root number: $i$ $|\Gal(K/\Q_{ 2 })|$: $12$ This field is 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: 2.4.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{4} - x + 1$$ Relative Eisenstein polynomial: $x^{3} + \left(2 t^{3} + 2 t^{2} + 2 t + 2\right) x^{2} + 2 \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_3:C_4$ (as 12T5) Inertia group: Intransitive group isomorphic to $C_3$ Unramified degree: $4$ Tame degree: $3$ Wild slopes: None Galois mean slope: $2/3$ Galois splitting model: $$x^{12} - 3 x^{11} - 888 x^{10} + 2660 x^{9} + 308151 x^{8} - 878220 x^{7} - 53209587 x^{6} + 132388596 x^{5} + 4800412971 x^{4} - 8975649010 x^{3} - 213443493450 x^{2} + 211625342025 x + 3644274033955$$