# Properties

 Label 5.14.0.1 Base $$\Q_{5}$$ Degree $$14$$ e $$1$$ f $$14$$ c $$0$$ Galois group $C_{14}$ (as 14T1)

# Related objects

## Defining polynomial

 $$x^{14} - x^{3} + x^{2} - 2 x + 2$$

## Invariants

 Base field: $\Q_{5}$ Degree $d$ : $14$ Ramification exponent $e$ : $1$ Residue field degree $f$ : $14$ Discriminant exponent $c$ : $0$ Discriminant root field: $\Q_{5}(\sqrt{*})$ Root number: $1$ $|\Gal(K/\Q_{ 5 })|$: $14$ This field is Galois and abelian over $\Q_{5}$.

## 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: 5.14.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of $$x^{14} - x^{3} + x^{2} - 2 x + 2$$ Relative Eisenstein polynomial: $x - 5 \in\Q_{5}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_{14}$ (as 14T1) Inertia group: Trivial Unramified degree: $14$ Tame degree: $1$ Wild slopes: None Galois mean slope: $0$ Galois splitting model: $x^{14} - x^{13} + 2 x^{12} + 24 x^{11} - 17 x^{10} + 27 x^{9} + 143 x^{8} - 81 x^{7} + 83 x^{6} + 209 x^{5} + 163 x^{4} - 88 x^{3} + 235 x^{2} + 168 x + 79$