# Properties

 Label 23.12.0.1 Base $$\Q_{23}$$ Degree $$12$$ e $$1$$ f $$12$$ c $$0$$ Galois group $C_{12}$ (as 12T1)

# Related objects

## Defining polynomial

 $$x^{12} + x^{2} - 3 x + 7$$

## Invariants

 Base field: $\Q_{23}$ Degree $d$ : $12$ Ramification exponent $e$ : $1$ Residue field degree $f$ : $12$ Discriminant exponent $c$ : $0$ Discriminant root field: $\Q_{23}(\sqrt{*})$ Root number: $1$ $|\Gal(K/\Q_{ 23 })|$: $12$ This field is Galois and abelian over $\Q_{23}$.

## 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: 23.12.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of $$x^{12} + x^{2} - 3 x + 7$$ Relative Eisenstein polynomial: $x - 23 \in\Q_{23}(t)[x]$

## 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: $x^{12} - x^{11} - 44 x^{10} + 23 x^{9} + 608 x^{8} - 288 x^{7} - 3367 x^{6} + 1647 x^{5} + 7459 x^{4} - 2633 x^{3} - 7037 x^{2} + 1034 x + 2209$