# Properties

 Label 2.6.4.1 Base $$\Q_{2}$$ Degree $$6$$ e $$3$$ f $$2$$ c $$4$$ Galois group $S_3$ (as 6T2)

# Related objects

## Defining polynomial

 $$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$ x^6 + 3*x^5 + 6*x^4 + 3*x^3 + 9*x + 9

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $6$ Ramification exponent $e$: $3$ Residue field degree $f$: $2$ Discriminant exponent $c$: $4$ Discriminant root field: $\Q_{2}(\sqrt{5})$ Root number: $1$ $\card{ \Gal(K/\Q_{ 2 }) }$: $6$ This field is Galois over $\Q_{2}.$ Visible slopes: None

## 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: $$x^{3} - 2$$ x^3 - 2 $\ \in\Q_{2}(t)[x]$ Indices of inseparability: $[0]$

## Invariants of the Galois closure

 Galois group: $S_3$ (as 6T2) Inertia group: Intransitive group isomorphic to $C_3$ Wild inertia group: $C_1$ Unramified degree: $2$ Tame degree: $3$ Wild slopes: None Galois mean slope: $2/3$ Galois splitting model: $x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$ x^6 + 3*x^5 + 6*x^4 + 3*x^3 + 9*x + 9