# Properties

 Label 7.15.0.1 Base $$\Q_{7}$$ Degree $$15$$ e $$1$$ f $$15$$ c $$0$$ Galois group $C_{15}$ (as 15T1)

# Related objects

## Defining polynomial

 $$x^{15} + 5 x^{6} + 6 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 2 x + 4$$ x^15 + 5*x^6 + 6*x^5 + 6*x^4 + 4*x^3 + x^2 + 2*x + 4

## Invariants

 Base field: $\Q_{7}$ Degree $d$: $15$ Ramification exponent $e$: $1$ Residue field degree $f$: $15$ Discriminant exponent $c$: $0$ Discriminant root field: $\Q_{7}$ Root number: $1$ $\card{ \Gal(K/\Q_{ 7 }) }$: $15$ This field is Galois and abelian over $\Q_{7}.$ Visible slopes: None

## 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: 7.15.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of $$x^{15} + 5 x^{6} + 6 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 2 x + 4$$ x^15 + 5*x^6 + 6*x^5 + 6*x^4 + 4*x^3 + x^2 + 2*x + 4 Relative Eisenstein polynomial: $$x - 7$$ x - 7 $\ \in\Q_{7}(t)[x]$

## Ramification polygon

The ramification polygon is trivial for unramified extensions.

## Invariants of the Galois closure

 Galois group: $C_{15}$ (as 15T1) Inertia group: trivial Wild inertia group: $C_1$ Unramified degree: $15$ Tame degree: $1$ Wild slopes: None Galois mean slope: $0$ Galois splitting model: Not computed