# Properties

 Label 2.6.9.8 Base $$\Q_{2}$$ Degree $$6$$ e $$2$$ f $$3$$ c $$9$$ Galois group $A_4\times C_2$ (as 6T6)

# Related objects

## Defining polynomial

 $$x^{6} - 8 x^{5} + 62 x^{4} + 224 x^{3} + 316 x^{2} + 1184 x + 1608$$ x^6 - 8*x^5 + 62*x^4 + 224*x^3 + 316*x^2 + 1184*x + 1608

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $6$ Ramification exponent $e$: $2$ Residue field degree $f$: $3$ Discriminant exponent $c$: $9$ Discriminant root field: $\Q_{2}(\sqrt{-2\cdot 5})$ Root number: $-i$ $\card{ \Aut(K/\Q_{ 2 }) }$: $2$ This field is not Galois over $\Q_{2}.$ Visible slopes: $[3]$

## 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: 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{3} + x + 1$$ x^3 + x + 1 Relative Eisenstein polynomial: $$x^{2} + \left(4 t^{2} + 4 t\right) x + 12 t^{2} + 4 t + 2$$ x^2 + (4*t^2 + 4*t)*x + 12*t^2 + 4*t + 2 $\ \in\Q_{2}(t)[x]$

## Ramification polygon

 Residual polynomials: $z + 1$ Associated inertia: $1$ Indices of inseparability: $[2, 0]$

## Invariants of the Galois closure

 Galois group: $C_2\times A_4$ (as 6T6) Inertia group: Intransitive group isomorphic to $C_2^3$ Wild inertia group: $C_2^3$ Unramified degree: $3$ Tame degree: $1$ Wild slopes: $[2, 2, 3]$ Galois mean slope: $9/4$ Galois splitting model: $x^{6} + 6 x^{4} - 24$