# Properties

 Label 13.12.0.1 Base $$\Q_{13}$$ Degree $$12$$ e $$1$$ f $$12$$ c $$0$$ Galois group $C_{12}$ (as 12T1)

# Related objects

## Defining polynomial

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

## Invariants

 Base field: $\Q_{13}$ Degree $d$ : $12$ Ramification exponent $e$ : $1$ Residue field degree $f$ : $12$ Discriminant exponent $c$ : $0$ Discriminant root field: $\Q_{13}(\sqrt{*})$ Root number: $1$ $|\Gal(K/\Q_{ 13 })|$: $12$ This field is Galois and abelian over $\Q_{13}$.

## 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: 13.12.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of $$x^{12} + x^{2} - x + 2$$ Relative Eisenstein polynomial: $x - 13 \in\Q_{13}(t)[x]$

## 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: $x^{12} + 3 x^{10} - x^{9} + 9 x^{8} + 9 x^{7} + 28 x^{6} + 18 x^{5} + 75 x^{4} + 26 x^{3} + 9 x^{2} + 3 x + 1$