# Properties

 Label 13.12.10.5 Base $$\Q_{13}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$10$$ Galois group $C_{12}$ (as 12T1)

# Related objects

## Defining polynomial

 $$x^{12} + 65 x^{6} + 1352$$

## Invariants

 Base field: $\Q_{13}$ Degree $d$ : $12$ Ramification exponent $e$ : $6$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $10$ Discriminant root field: $\Q_{13}(\sqrt{*})$ 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: $\Q_{13}(\sqrt{*})$ $\cong \Q_{13}(t)$ where $t$ is a root of $$x^{2} - x + 2$$ Relative Eisenstein polynomial: $x^{6} - 13 t^{3} \in\Q_{13}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_{12}$ (as 12T1) Inertia group: Intransitive group isomorphic to $C_6$ Unramified degree: $2$ Tame degree: $6$ Wild slopes: None Galois mean slope: $5/6$ Galois splitting model: $x^{12} - x^{11} + 10 x^{10} - 15 x^{9} + x^{8} - 55 x^{7} + 24 x^{6} + 155 x^{5} + 471 x^{4} + 505 x^{3} + 415 x^{2} + 309 x + 521$