# Properties

 Label 2.6.11.9 Base $$\Q_{2}$$ Degree $$6$$ e $$6$$ f $$1$$ c $$11$$ Galois group $D_{6}$ (as 6T3)

# Related objects

## Defining polynomial

 $$x^{6} + 4 x^{5} + 4 x^{2} + 2$$ x^6 + 4*x^5 + 4*x^2 + 2

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $6$ Ramification exponent $e$: $6$ Residue field degree $f$: $1$ Discriminant exponent $c$: $11$ Discriminant root field: $\Q_{2}(\sqrt{-2})$ 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: $\Q_{2}$ Relative Eisenstein polynomial: $$x^{6} + 4 x^{5} + 4 x^{2} + 2$$ x^6 + 4*x^5 + 4*x^2 + 2

## Ramification polygon

 Residual polynomials: $z + 1$,$z^{4} + z^{2} + 1$ Associated inertia: $1$,$2$ Indices of inseparability: $[6, 0]$

## Invariants of the Galois closure

 Galois group: $D_6$ (as 6T3) Inertia group: $C_6$ (as 6T1) Wild inertia group: $C_2$ Unramified degree: $2$ Tame degree: $3$ Wild slopes: $[3]$ Galois mean slope: $11/6$ Galois splitting model: $x^{6} + 6 x^{4} + 2$