# Properties

 Label 13.8.6.1 Base $$\Q_{13}$$ Degree $$8$$ e $$4$$ f $$2$$ c $$6$$ Galois group $C_4\times C_2$ (as 8T2)

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## Defining polynomial

 $$x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24242 x^{4} + 15024 x^{3} + 14408 x^{2} + 86496 x + 254881$$ x^8 + 48*x^7 + 872*x^6 + 7200*x^5 + 24242*x^4 + 15024*x^3 + 14408*x^2 + 86496*x + 254881

## Invariants

 Base field: $\Q_{13}$ Degree $d$: $8$ Ramification exponent $e$: $4$ Residue field degree $f$: $2$ Discriminant exponent $c$: $6$ Discriminant root field: $\Q_{13}$ Root number: $-1$ $\card{ \Gal(K/\Q_{ 13 }) }$: $8$ This field is Galois and abelian over $\Q_{13}.$ Visible slopes: None

## 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: $\Q_{13}(\sqrt{2})$ $\cong \Q_{13}(t)$ where $t$ is a root of $$x^{2} + 12 x + 2$$ x^2 + 12*x + 2 Relative Eisenstein polynomial: $$x^{4} + 13$$ x^4 + 13 $\ \in\Q_{13}(t)[x]$

## Ramification polygon

 Residual polynomials: $z^{3} + 4z^{2} + 6z + 4$ Associated inertia: $1$ Indices of inseparability: $[0]$

## Invariants of the Galois closure

 Galois group: $C_2\times C_4$ (as 8T2) Inertia group: Intransitive group isomorphic to $C_4$ Wild inertia group: $C_1$ Unramified degree: $2$ Tame degree: $4$ Wild slopes: None Galois mean slope: $3/4$ Galois splitting model: $x^{8} - x^{7} + 5 x^{6} + 18 x^{5} + 37 x^{4} + 20 x^{3} + 2 x^{2} - 12 x + 9$