# Properties

 Label 13.6.5.5 Base $$\Q_{13}$$ Degree $$6$$ e $$6$$ f $$1$$ c $$5$$ Galois group $C_6$ (as 6T1)

# Related objects

## Defining polynomial

 $$x^{6} + 65$$ x^6 + 65

## Invariants

 Base field: $\Q_{13}$ Degree $d$: $6$ Ramification exponent $e$: $6$ Residue field degree $f$: $1$ Discriminant exponent $c$: $5$ Discriminant root field: $\Q_{13}(\sqrt{13\cdot 2})$ Root number: $-1$ $\card{ \Gal(K/\Q_{ 13 }) }$: $6$ 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: $\Q_{13}$ Relative Eisenstein polynomial: $$x^{6} + 65$$ x^6 + 65

## Ramification polygon

 Residual polynomials: $z^{5} + 6z^{4} + 2z^{3} + 7z^{2} + 2z + 6$ Associated inertia: $1$ Indices of inseparability: $[0]$

## Invariants of the Galois closure

 Galois group: $C_6$ (as 6T1) Inertia group: $C_6$ (as 6T1) Wild inertia group: $C_1$ Unramified degree: $1$ Tame degree: $6$ Wild slopes: None Galois mean slope: $5/6$ Galois splitting model: $x^{6} - x^{5} - 18 x^{4} + 17 x^{3} + 58 x^{2} - 16 x - 1$