# Properties

 Label 2.10.15.11 Base $$\Q_{2}$$ Degree $$10$$ e $$2$$ f $$5$$ c $$15$$ Galois group $C_2 \times (C_2^4 : C_5)$ (as 10T14)

# Related objects

## Defining polynomial

 $$x^{10} + 2 x^{8} - 8 x^{6} - 16 x^{4} + 16 x^{2} + 1056$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$ : $10$ Ramification exponent $e$ : $2$ Residue field degree $f$ : $5$ Discriminant exponent $c$ : $15$ Discriminant root field: $\Q_{2}(\sqrt{-2})$ 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: 2.5.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{5} + x^{2} + 1$$ Relative Eisenstein polynomial: $x^{2} - 4 t^{3} - 2 \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_2\times C_2^4:C_5$ (as 10T14) Inertia group: Intransitive group isomorphic to $C_2^5$ Unramified degree: $5$ Tame degree: $1$ Wild slopes: [2, 2, 2, 2, 3] Galois mean slope: $39/16$ Galois splitting model: $x^{10} + 8 x^{8} + 8 x^{6} - 40 x^{4} - 32 x^{2} + 32$