# Properties

 Label 2.8.8.2 Base $$\Q_{2}$$ Degree $$8$$ e $$2$$ f $$4$$ c $$8$$ Galois group $C_2^2:C_4$ (as 8T10)

# Related objects

## Defining polynomial

 $$x^{8} + 2 x^{7} + 8 x^{2} + 48$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $8$ Ramification exponent $e$: $2$ Residue field degree $f$: $4$ Discriminant exponent $c$: $8$ Discriminant root field: $\Q_{2}$ Root number: $-1$ $|\Aut(K/\Q_{ 2 })|$: $4$ 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: 2.4.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{4} - x + 1$$ Relative Eisenstein polynomial: $$x^{2} + \left(6 t^{3} + 4 t^{2} + 2 t\right) x + 4 t^{3} + 2 t^{2} + 6$$$\ \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_2^2:C_4$ (as 8T10) Inertia group: Intransitive group isomorphic to $C_2^2$ Wild inertia group: $C_2^2$ Unramified degree: $4$ Tame degree: $1$ Wild slopes: [2, 2] Galois mean slope: $3/2$ Galois splitting model: $x^{8} - 2 x^{6} + 4 x^{4} - 3 x^{2} + 1$