# Properties

 Label 2.10.16.10 Base $$\Q_{2}$$ Degree $$10$$ e $$10$$ f $$1$$ c $$16$$ Galois group $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29)

# Related objects

## Defining polynomial

 $$x^{10} - 2 x^{9} + 2 x^{7} + 2 x^{6} - 6 x^{4} - 4 x^{3} - 6 x^{2} - 2$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$ : $10$ Ramification exponent $e$ : $10$ Residue field degree $f$ : $1$ Discriminant exponent $c$ : $16$ Discriminant root field: $\Q_{2}(\sqrt{-1})$ Root number: $-i$ $|\Aut(K/\Q_{ 2 })|$: $2$ This field is not Galois over $\Q_{2}$.

## 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: $\Q_{2}$ Relative Eisenstein polynomial: $$x^{10} - 2 x^{9} + 2 x^{7} + 2 x^{6} - 6 x^{4} - 4 x^{3} - 6 x^{2} - 2$$

## Invariants of the Galois closure

 Galois group: $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29) Inertia group: $C_2 \times (C_2^4 : C_5)$ Unramified degree: $4$ Tame degree: $5$ Wild slopes: [2, 12/5, 12/5, 12/5, 12/5] Galois mean slope: $187/80$ Galois splitting model: $x^{10} + 2 x^{8} - 4 x^{6} + 8 x^{2} + 4$