# Properties

 Label 2.9.6.1 Base $$\Q_{2}$$ Degree $$9$$ e $$3$$ f $$3$$ c $$6$$ Galois group $S_3\times C_3$ (as 9T4)

# Related objects

## Defining polynomial

 $$x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$$ x^9 + 3*x^7 + 9*x^6 + 3*x^5 - 26*x^3 + 9*x^2 - 27*x + 29

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $9$ Ramification exponent $e$: $3$ Residue field degree $f$: $3$ Discriminant exponent $c$: $6$ Discriminant root field: $\Q_{2}(\sqrt{5})$ Root number: $1$ $\card{ \Aut(K/\Q_{ 2 }) }$: $3$ This field is not Galois over $\Q_{2}.$ Visible slopes: None

## 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: $$x^{3} + 2$$ x^3 + 2 $\ \in\Q_{2}(t)[x]$

## Ramification polygon

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

## Invariants of the Galois closure

 Galois group: $C_3\times S_3$ (as 9T4) Inertia group: Intransitive group isomorphic to $C_3$ Wild inertia group: $C_1$ Unramified degree: $6$ Tame degree: $3$ Wild slopes: None Galois mean slope: $2/3$ Galois splitting model: $x^{9} - 3 x^{7} - x^{6} + 3 x^{5} + 2 x^{4} + 4 x^{3} - x^{2} - 5 x + 1$