# Properties

 Label 13.12.9.4 Base $$\Q_{13}$$ Degree $$12$$ e $$4$$ f $$3$$ c $$9$$ Galois group $C_{12}$ (as 12T1)

# Related objects

## Defining polynomial

 $$x^{12} + 234 x^{8} + 16900 x^{4} + 474552$$

## Invariants

 Base field: $\Q_{13}$ Degree $d$ : $12$ Ramification exponent $e$ : $4$ Residue field degree $f$ : $3$ Discriminant exponent $c$ : $9$ Discriminant root field: $\Q_{13}(\sqrt{13*})$ 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: 13.3.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of $$x^{3} - 2 x + 6$$ Relative Eisenstein polynomial: $x^{4} - 13 t^{3} \in\Q_{13}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_{12}$ (as 12T1) Inertia group: Intransitive group isomorphic to $C_4$ Unramified degree: $3$ Tame degree: $4$ Wild slopes: None Galois mean slope: $3/4$ Galois splitting model: $x^{12} - 3 x^{11} - 81 x^{10} + 181 x^{9} + 2148 x^{8} - 3324 x^{7} - 22545 x^{6} + 18504 x^{5} + 94026 x^{4} - 19578 x^{3} - 107604 x^{2} + 1035 x + 26711$