# Properties

 Label 2.14.0.1 Base $$\Q_{2}$$ Degree $$14$$ e $$1$$ f $$14$$ c $$0$$ Galois group $C_{14}$ (as 14T1)

# Related objects

## Defining polynomial

 $$x^{14} - x^{5} - x^{3} - x + 1$$

## Invariants

 Base field: $\Q_{2}$ Degree $d$: $14$ Ramification exponent $e$: $1$ Residue field degree $f$: $14$ Discriminant exponent $c$: $0$ Discriminant root field: $\Q_{2}(\sqrt{*})$ 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: 2.14.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of $$x^{14} - x^{5} - x^{3} - x + 1$$ Relative Eisenstein polynomial: $x - 2 \in\Q_{2}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_{14}$ (as 14T1) Inertia group: trivial Unramified degree: $14$ Tame degree: $1$ Wild slopes: None Galois mean slope: $0$ Galois splitting model: $x^{14} + 21 x^{12} - 42 x^{11} + 350 x^{10} - 553 x^{9} + 2184 x^{8} - 2696 x^{7} + 8869 x^{6} - 8701 x^{5} + 18151 x^{4} - 8246 x^{3} + 17920 x^{2} - 8148 x + 9409$