# Properties

 Label 2.8.18.16 Base $$\Q_{2}$$ Degree $$8$$ e $$4$$ f $$2$$ c $$18$$ Galois group $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29)

# Related objects

## Defining polynomial

 $$x^{8} + 6 x^{6} + 4 x^{2} + 4$$ x^8 + 6*x^6 + 4*x^2 + 4

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $8$ Ramification exponent $e$: $4$ Residue field degree $f$: $2$ Discriminant exponent $c$: $18$ Discriminant root field: $\Q_{2}$ Root number: $1$ $\card{ \Aut(K/\Q_{ 2 }) }$: $2$ This field is not Galois over $\Q_{2}.$ Visible slopes: $[2,7/2]$

## 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: $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{2} - x + 1$$ x^2 - x + 1 Relative Eisenstein polynomial: $$x^{4} + \left(10 t + 6\right) x^{2} + 6$$ x^4 + (10*t + 6)*x^2 + 6 $\ \in\Q_{2}(t)[x]$ Indices of inseparability: $[6, 2, 0]$

## Invariants of the Galois closure

 Galois group: $C_2\wr C_2^2$ (as 8T29) Inertia group: Intransitive group isomorphic to $C_2^2\wr C_2$ Wild inertia group: $C_2^2\wr C_2$ Unramified degree: $2$ Tame degree: $1$ Wild slopes: $[2, 2, 3, 7/2, 7/2]$ Galois mean slope: $51/16$ Galois splitting model: $x^{8} + 2 x^{6} - 26 x^{4} - 72 x^{2} + 36$