# Properties

 Label 13.12.8.1 Base $$\Q_{13}$$ Degree $$12$$ e $$3$$ f $$4$$ c $$8$$ Galois group $C_{12}$ (as 12T1)

# Related objects

## Defining polynomial

 $$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$

## Invariants

 Base field: $\Q_{13}$ Degree $d$ : $12$ Ramification exponent $e$ : $3$ Residue field degree $f$ : $4$ Discriminant exponent $c$ : $8$ Discriminant root field: $\Q_{13}(\sqrt{*})$ Root number: $1$ $|\Gal(K/\Q_{ 13 })|$: $12$ 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: 13.4.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of $$x^{4} + x^{2} - x + 2$$ Relative Eisenstein polynomial: $x^{3} - 13 t^{3} \in\Q_{13}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_{12}$ (as 12T1) Inertia group: Intransitive group isomorphic to $C_3$ Unramified degree: $4$ Tame degree: $3$ Wild slopes: None Galois mean slope: $2/3$ Galois splitting model: $x^{12} - x^{11} + 5 x^{10} - 10 x^{9} + 31 x^{8} + 50 x^{7} + 84 x^{6} + 85 x^{5} + 201 x^{4} + 55 x^{3} + 15 x^{2} + 4 x + 1$