# Properties

 Label 2.6.9.3 Base $$\Q_{2}$$ Degree $$6$$ e $$2$$ f $$3$$ c $$9$$ Galois group $C_6$ (as 6T1)

# Related objects

## Defining polynomial

 $$x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$$ x^6 + 12*x^5 + 86*x^4 + 352*x^3 + 892*x^2 + 1552*x + 1384

## 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: $-1$ $\card{ \Gal(K/\Q_{ 2 }) }$: $6$ This field is Galois and abelian 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: $$x^{2} + \left(4 t + 4\right) x + 4 t^{2} + 8 t + 10$$ x^2 + (4*t + 4)*x + 4*t^2 + 8*t + 10 $\ \in\Q_{2}(t)[x]$

## Ramification polygon

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

## Invariants of the Galois closure

 Galois group: $C_6$ (as 6T1) Inertia group: Intransitive group isomorphic to $C_2$ Wild inertia group: $C_2$ Unramified degree: $3$ Tame degree: $1$ Wild slopes: $[3]$ Galois mean slope: $3/2$ Galois splitting model: $x^{6} + 12 x^{4} + 36 x^{2} + 24$