# Properties

 Label 19.12.0.1 Base $$\Q_{19}$$ Degree $$12$$ e $$1$$ f $$12$$ c $$0$$ Galois group $C_{12}$ (as 12T1)

# Related objects

## Defining polynomial

 $$x^{12} + 3 x^{7} + 2 x^{6} + 18 x^{5} + 2 x^{4} + 9 x^{3} + 16 x^{2} + 7 x + 2$$ x^12 + 3*x^7 + 2*x^6 + 18*x^5 + 2*x^4 + 9*x^3 + 16*x^2 + 7*x + 2

## Invariants

 Base field: $\Q_{19}$ Degree $d$: $12$ Ramification exponent $e$: $1$ Residue field degree $f$: $12$ Discriminant exponent $c$: $0$ Discriminant root field: $\Q_{19}(\sqrt{2})$ Root number: $1$ $\card{ \Gal(K/\Q_{ 19 }) }$: $12$ 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: 19.12.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of $$x^{12} + 3 x^{7} + 2 x^{6} + 18 x^{5} + 2 x^{4} + 9 x^{3} + 16 x^{2} + 7 x + 2$$ x^12 + 3*x^7 + 2*x^6 + 18*x^5 + 2*x^4 + 9*x^3 + 16*x^2 + 7*x + 2 Relative Eisenstein polynomial: $$x - 19$$ x - 19 $\ \in\Q_{19}(t)[x]$

## Ramification polygon

The ramification polygon is trivial for unramified extensions.