# Properties

 Label 13.12.10.1 Base $$\Q_{13}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$10$$ Galois group $C_6\times C_2$ (as 12T2)

# Related objects

## Defining polynomial

 $$x^{12} - 117 x^{6} + 10816$$

## Invariants

 Base field: $\Q_{13}$ Degree $d$ : $12$ Ramification exponent $e$ : $6$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $10$ Discriminant root field: $\Q_{13}$ 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: $\Q_{13}(\sqrt{*})$ $\cong \Q_{13}(t)$ where $t$ is a root of $$x^{2} - x + 2$$ Relative Eisenstein polynomial: $x^{6} - 13 t^{6} \in\Q_{13}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_2\times C_6$ (as 12T2) Inertia group: Intransitive group isomorphic to $C_6$ Unramified degree: $2$ Tame degree: $6$ Wild slopes: None Galois mean slope: $5/6$ Galois splitting model: $x^{12} - x^{11} - 16 x^{10} + 11 x^{9} + 79 x^{8} - 29 x^{7} - 145 x^{6} + 25 x^{5} + 107 x^{4} - 2 x^{3} - 27 x^{2} - 3 x + 1$