# Properties

 Label 2.14.21.34 Base $$\Q_{2}$$ Degree $$14$$ e $$2$$ f $$7$$ c $$21$$ Galois group $C_{14}$ (as 14T1)

# Related objects

## Defining polynomial

 $$x^{14} + 4 x^{13} + 8 x^{12} + 4 x^{11} + 5 x^{10} + 8 x^{9} - 6 x^{8} - 6 x^{7} + x^{6} + 6 x^{5} + 2 x^{3} + 7 x^{2} + 6 x - 7$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $14$ Ramification exponent $e$: $2$ Residue field degree $f$: $7$ Discriminant exponent $c$: $21$ Discriminant root field: $\Q_{2}(\sqrt{2})$ Root number: $1$ $|\Gal(K/\Q_{ 2 })|$: $14$ This field is Galois and abelian 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: 2.7.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{7} - x + 1$$ Relative Eisenstein polynomial: $$x^{2} + 4 t^{3} x + 2 t^{6} + 6 t^{5} + 6 t^{4} + 2 t^{3} + 2 t^{2} + 2 t$$$\ \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_{14}$ (as 14T1) Inertia group: Intransitive group isomorphic to $C_2$ Unramified degree: $7$ Tame degree: $1$ Wild slopes: [3] Galois mean slope: $3/2$ Galois splitting model: $x^{14} + 84 x^{12} + 2156 x^{10} + 19208 x^{8} + 60368 x^{6} + 47040 x^{4} + 12544 x^{2} + 896$