# Properties

 Label 17.12.0.1 Base $$\Q_{17}$$ Degree $$12$$ e $$1$$ f $$12$$ c $$0$$ Galois group $C_{12}$ (as 12T1)

# Related objects

## Defining polynomial

 $$x^{12} + 3 x^{2} - 2 x + 5$$

## Invariants

 Base field: $\Q_{17}$ Degree $d$ : $12$ Ramification exponent $e$ : $1$ Residue field degree $f$ : $12$ Discriminant exponent $c$ : $0$ Discriminant root field: $\Q_{17}(\sqrt{*})$ Root number: $1$ $|\Gal(K/\Q_{ 17 })|$: $12$ This field is Galois and abelian over $\Q_{17}$.

## 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: 17.12.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of $$x^{12} + 3 x^{2} - 2 x + 5$$ Relative Eisenstein polynomial: $x - 17 \in\Q_{17}(t)[x]$

## 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: $x^{12} - x^{11} + 2 x^{10} + 20 x^{9} - 13 x^{8} + 19 x^{7} + 85 x^{6} - 51 x^{5} + 94 x^{4} + 2 x^{3} - 13 x^{2} + 77 x + 47$