# Properties

 Label 19.8.4.1 Base $$\Q_{19}$$ Degree $$8$$ e $$2$$ f $$4$$ c $$4$$ Galois group $C_4\times C_2$ (as 8T2)

# Related objects

## Defining polynomial

 $$x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$$ x^8 + 80*x^6 + 22*x^5 + 2250*x^4 - 792*x^3 + 25817*x^2 - 22946*x + 107924

## Invariants

 Base field: $\Q_{19}$ Degree $d$: $8$ Ramification exponent $e$: $2$ Residue field degree $f$: $4$ Discriminant exponent $c$: $4$ Discriminant root field: $\Q_{19}$ Root number: $-1$ $\card{ \Gal(K/\Q_{ 19 }) }$: $8$ This field is Galois and abelian over $\Q_{19}.$ 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: 19.4.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of $$x^{4} + 2 x^{2} + 11 x + 2$$ x^4 + 2*x^2 + 11*x + 2 Relative Eisenstein polynomial: $$x^{2} + 19$$ x^2 + 19 $\ \in\Q_{19}(t)[x]$ Indices of inseparability: $[0]$

## Invariants of the Galois closure

 Galois group: $C_2\times C_4$ (as 8T2) Inertia group: Intransitive group isomorphic to $C_2$ Wild inertia group: $C_1$ Unramified degree: $4$ Tame degree: $2$ Wild slopes: None Galois mean slope: $1/2$ Galois splitting model: Not computed