# Properties

 Label 7.6.3.1 Base $$\Q_{7}$$ Degree $$6$$ e $$2$$ f $$3$$ c $$3$$ Galois group $C_6$ (as 6T1)

# Related objects

## Defining polynomial

 $$x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$$ x^6 + 861*x^5 + 33033*x^4 + 1385475*x^3 + 277830*x^2 + 8232*x - 1372

## Invariants

 Base field: $\Q_{7}$ Degree $d$: $6$ Ramification exponent $e$: $2$ Residue field degree $f$: $3$ Discriminant exponent $c$: $3$ Discriminant root field: $\Q_{7}(\sqrt{7})$ Root number: $i$ $\card{ \Gal(K/\Q_{ 7 }) }$: $6$ 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: 7.3.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of $$x^{3} + 6 x^{2} + 4$$ x^3 + 6*x^2 + 4 Relative Eisenstein polynomial: $$x^{2} + \left(28 t^{2} + 42 t + 35\right) x + 7 t$$ x^2 + (28*t^2 + 42*t + 35)*x + 7*t $\ \in\Q_{7}(t)[x]$

## Ramification polygon

 Residual polynomials: $z + 2$ Associated inertia: $1$ Indices of inseparability: $[0]$

## Invariants of the Galois closure

 Galois group: $C_6$ (as 6T1) Inertia group: Intransitive group isomorphic to $C_2$ Wild inertia group: $C_1$ Unramified degree: $3$ Tame degree: $2$ Wild slopes: None Galois mean slope: $1/2$ Galois splitting model: $x^{6} - x^{5} + 21 x^{4} - 22 x^{3} + 58 x^{2} + 23 x + 155$