# Properties

 Label 13.8.6.4 Base $$\Q_{13}$$ Degree $$8$$ e $$4$$ f $$2$$ c $$6$$ Galois group $C_8$ (as 8T1)

# Learn more about

## Defining polynomial

 $$x^{8} - 13 x^{4} + 338$$

## Invariants

 Base field: $\Q_{13}$ Degree $d$ : $8$ Ramification exponent $e$ : $4$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $6$ Discriminant root field: $\Q_{13}(\sqrt{*})$ Root number: $1$ $|\Gal(K/\Q_{ 13 })|$: $8$ 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: $\Q_{13}(\sqrt{*})$ $\cong \Q_{13}(t)$ where $t$ is a root of $$x^{2} - x + 2$$ Relative Eisenstein polynomial: $x^{4} - 13 t \in\Q_{13}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_8$ (as 8T1) Inertia group: Intransitive group isomorphic to $C_4$ Unramified degree: $2$ Tame degree: $4$ Wild slopes: None Galois mean slope: $3/4$ Galois splitting model: $x^{8} - x^{7} - 202 x^{6} - 298 x^{5} + 11401 x^{4} + 35231 x^{3} - 125184 x^{2} - 387700 x + 204608$