# Properties

 Label 5.13.0.1 Base $$\Q_{5}$$ Degree $$13$$ e $$1$$ f $$13$$ c $$0$$ Galois group $C_{13}$ (as 13T1)

# Related objects

## Defining polynomial

 $$x^{13} + x^{2} - 2 x + 2$$

## Invariants

 Base field: $\Q_{5}$ Degree $d$ : $13$ Ramification exponent $e$ : $1$ Residue field degree $f$ : $13$ Discriminant exponent $c$ : $0$ Discriminant root field: $\Q_{5}$ Root number: $1$ $|\Gal(K/\Q_{ 5 })|$: $13$ This field is Galois and abelian over $\Q_{5}$.

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$.

## Unramified/totally ramified tower

 Unramified subfield: 5.13.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of $$x^{13} + x^{2} - 2 x + 2$$ Relative Eisenstein polynomial: $x - 5 \in\Q_{5}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_{13}$ (as 13T1) Inertia group: Trivial Unramified degree: $13$ Tame degree: $1$ Wild slopes: None Galois mean slope: $0$ Galois splitting model: $x^{13} - x^{12} - 24 x^{11} + 19 x^{10} + 190 x^{9} - 116 x^{8} - 601 x^{7} + 246 x^{6} + 738 x^{5} - 215 x^{4} - 291 x^{3} + 68 x^{2} + 10 x - 1$