# Properties

 Label 3.12.16.14 Base $$\Q_{3}$$ Degree $$12$$ e $$3$$ f $$4$$ c $$16$$ Galois group $C_{12}$ (as 12T1)

# Related objects

## Defining polynomial

 $$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$ x^12 + 72*x^11 - 36*x^10 + 108*x^9 - 108*x^8 + 54*x^7 + 72*x^6 - 81*x^5 - 81*x^4 - 81*x^3 + 81*x^2 - 81

## Invariants

 Base field: $\Q_{3}$ Degree $d$: $12$ Ramification exponent $e$: $3$ Residue field degree $f$: $4$ Discriminant exponent $c$: $16$ Discriminant root field: $\Q_{3}(\sqrt{2})$ Root number: $1$ $\card{ \Gal(K/\Q_{ 3 }) }$: $12$ This field is Galois and abelian over $\Q_{3}.$ Visible slopes: $[2]$

## 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: 3.4.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of $$x^{4} - x + 2$$ x^4 - x + 2 Relative Eisenstein polynomial: $$x^{3} + \left(-3 t^{3} + 12 t^{2} + 12 t - 3\right) x^{2} + \left(-9 t^{3} - 9 t + 9\right) x - 3 t^{3} + 12 t^{2} + 3 t - 6$$ x^3 + (-3*t^3 + 12*t^2 + 12*t - 3)*x^2 + (-9*t^3 - 9*t + 9)*x - 3*t^3 + 12*t^2 + 3*t - 6 $\ \in\Q_{3}(t)[x]$ Indices of inseparability: $[2, 0]$

## Invariants of the Galois closure

 Galois group: $C_{12}$ (as 12T1) Inertia group: Intransitive group isomorphic to $C_3$ Wild inertia group: $C_3$ Unramified degree: $4$ Tame degree: $1$ Wild slopes: $[2]$ Galois mean slope: $4/3$ Galois splitting model: $x^{12} + 3 x^{10} - x^{9} + 9 x^{8} + 9 x^{7} + 28 x^{6} + 18 x^{5} + 75 x^{4} + 26 x^{3} + 9 x^{2} + 3 x + 1$