# Properties

 Label 2.8.12.15 Base $$\Q_{2}$$ Degree $$8$$ e $$4$$ f $$2$$ c $$12$$ Galois group $C_2^2:C_4$ (as 8T10)

# Related objects

## Defining polynomial

 $$x^{8} + 2 x^{7} + 2 x^{4} + 12$$

## Invariants

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

## Invariants of the Galois closure

 Galois group: $C_2^2:C_4$ (as 8T10) Inertia group: Intransitive group isomorphic to $C_2^2$ Unramified degree: $4$ Tame degree: $1$ Wild slopes: [2, 2] Galois mean slope: $3/2$ Galois splitting model: $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 14 x^{4} - 8 x^{3} - 16 x^{2} + 16 x - 4$