# Properties

 Label 3.12.0.1 Base $$\Q_{3}$$ Degree $$12$$ e $$1$$ f $$12$$ c $$0$$ Galois group $C_{12}$ (as 12T1)

# Learn more about

## Defining polynomial

 $$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$ ## Invariants

 Base field: $\Q_{3}$ Degree $d$: $12$ Ramification exponent $e$: $1$ Residue field degree $f$: $12$ Discriminant exponent $c$: $0$ Discriminant root field: $\Q_{3}(\sqrt{2})$ Root number: $1$ $|\Gal(K/\Q_{ 3 })|$: $12$ This field is Galois and abelian over $\Q_{3}.$

## 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: 3.12.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of $$x^{12} + 2 x^{4} - x^{3} + 2 x^{2} - 2 x + 2$$ Relative Eisenstein polynomial: $$x - 3$$$\ \in\Q_{3}(t)[x]$ ## Invariants of the Galois closure

 Galois group: $C_{12}$ (as 12T1) Inertia group: trivial Unramified degree: $12$ Tame degree: $1$ Wild slopes: None Galois mean slope: $0$ Galois splitting model: $x^{12} - x^{11} + 3 x^{10} - 4 x^{9} + 9 x^{8} + 2 x^{7} + 12 x^{6} + x^{5} + 25 x^{4} - 11 x^{3} + 5 x^{2} - 2 x + 1$