# Properties

 Label 2.6.6.7 Base $$\Q_{2}$$ Degree $$6$$ e $$6$$ f $$1$$ c $$6$$ Galois group $S_4$ (as 6T7)

# Related objects

## Defining polynomial

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

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $6$ Ramification exponent $e$: $6$ Residue field degree $f$: $1$ Discriminant exponent $c$: $6$ Discriminant root field: $\Q_{2}$ Root number: $1$ $\card{ \Aut(K/\Q_{ 2 }) }$: $2$ This field is not Galois over $\Q_{2}.$ Visible slopes: $[4/3]$

## 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: $\Q_{2}$ Relative Eisenstein polynomial: $$x^{6} + 2 x^{2} + 2 x + 2$$ x^6 + 2*x^2 + 2*x + 2

## Ramification polygon

 Residual polynomials: $z + 1$,$z^{4} + z^{2} + 1$ Associated inertia: $1$,$2$ Indices of inseparability: $[1, 0]$

## Invariants of the Galois closure

 Galois group: $S_4$ (as 6T7) Inertia group: $A_4$ (as 6T4) Wild inertia group: $C_2^2$ Unramified degree: $2$ Tame degree: $3$ Wild slopes: $[4/3, 4/3]$ Galois mean slope: $7/6$ Galois splitting model: $x^{6} - 3 x^{4} - 6 x^{3} + 3 x^{2} - 1$