# Properties

 Label 7.7.12.1 Base $$\Q_{7}$$ Degree $$7$$ e $$7$$ f $$1$$ c $$12$$ Galois group $C_7$ (as 7T1)

# Related objects

## Defining polynomial

 $$x^{7} + 42 x^{6} + 7$$ x^7 + 42*x^6 + 7

## Invariants

 Base field: $\Q_{7}$ Degree $d$: $7$ Ramification exponent $e$: $7$ Residue field degree $f$: $1$ Discriminant exponent $c$: $12$ Discriminant root field: $\Q_{7}$ Root number: $1$ $\card{ \Gal(K/\Q_{ 7 }) }$: $7$ This field is Galois and abelian over $\Q_{7}.$ Visible slopes: $[2]$

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$.

## Unramified/totally ramified tower

 Unramified subfield: $\Q_{7}$ Relative Eisenstein polynomial: $$x^{7} + 42 x^{6} + 7$$ x^7 + 42*x^6 + 7

## Ramification polygon

 Residual polynomials: $z^{6} + 6$ Associated inertia: $1$ Indices of inseparability: $[6, 0]$

## Invariants of the Galois closure

 Galois group: $C_7$ (as 7T1) Inertia group: $C_7$ (as 7T1) Wild inertia group: $C_7$ Unramified degree: $1$ Tame degree: $1$ Wild slopes: $[2]$ Galois mean slope: $12/7$ Galois splitting model: $x^{7} - 21 x^{5} - 21 x^{4} + 91 x^{3} + 112 x^{2} - 84 x - 97$