# Properties

 Label 3.6.7.1 Base $$\Q_{3}$$ Degree $$6$$ e $$6$$ f $$1$$ c $$7$$ Galois group $S_3$ (as 6T2)

# Related objects

## Defining polynomial

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

## Invariants

 Base field: $\Q_{3}$ Degree $d$: $6$ Ramification exponent $e$: $6$ Residue field degree $f$: $1$ Discriminant exponent $c$: $7$ Discriminant root field: $\Q_{3}(\sqrt{3})$ Root number: $i$ $\card{ \Gal(K/\Q_{ 3 }) }$: $6$ This field is Galois over $\Q_{3}.$ Visible slopes: $[3/2]$

## 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: $\Q_{3}$ Relative Eisenstein polynomial: $$x^{6} + 6 x^{2} + 6$$ x^6 + 6*x^2 + 6

## Ramification polygon

 Residual polynomials: $2z^{2} + 1$,$z^{3} + 2$ Associated inertia: $1$,$1$ Indices of inseparability: $[2, 0]$

## Invariants of the Galois closure

 Galois group: $S_3$ (as 6T2) Inertia group: $S_3$ (as 6T2) Wild inertia group: $C_3$ Unramified degree: $1$ Tame degree: $2$ Wild slopes: $[3/2]$ Galois mean slope: $7/6$ Galois splitting model: $x^{6} - 3 x^{5} + 18 x^{4} - 31 x^{3} + 75 x^{2} - 60 x + 16$