# Properties

 Label 7.10.5.2 Base $$\Q_{7}$$ Degree $$10$$ e $$2$$ f $$5$$ c $$5$$ Galois group $C_{10}$ (as 10T1)

# Related objects

## Defining polynomial

 $$x^{10} - 2401 x^{2} + 67228$$

## Invariants

 Base field: $\Q_{7}$ Degree $d$ : $10$ Ramification exponent $e$ : $2$ Residue field degree $f$ : $5$ Discriminant exponent $c$ : $5$ Discriminant root field: $\Q_{7}(\sqrt{7*})$ Root number: $i$ $|\Gal(K/\Q_{ 7 })|$: $10$ This field is Galois and abelian over $\Q_{7}$.

## 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: 7.5.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of $$x^{5} - x + 4$$ Relative Eisenstein polynomial: $x^{2} - 7 t \in\Q_{7}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_{10}$ (as 10T1) Inertia group: Intransitive group isomorphic to $C_2$ Unramified degree: $5$ Tame degree: $2$ Wild slopes: None Galois mean slope: $1/2$ Galois splitting model: $x^{10} - x^{9} + 14 x^{8} - 7 x^{7} + 85 x^{6} - 29 x^{5} + 218 x^{4} - 8 x^{3} + 216 x^{2} - 48 x + 32$