# Properties

 Label 13.7.6.1 Base $$\Q_{13}$$ Degree $$7$$ e $$7$$ f $$1$$ c $$6$$ Galois group $D_{7}$ (as 7T2)

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## Defining polynomial

 $$x^{7} + 13$$ x^7 + 13

## Invariants

 Base field: $\Q_{13}$ Degree $d$: $7$ Ramification exponent $e$: $7$ Residue field degree $f$: $1$ Discriminant exponent $c$: $6$ Discriminant root field: $\Q_{13}(\sqrt{2})$ Root number: $1$ $\card{ \Aut(K/\Q_{ 13 }) }$: $1$ This field is not Galois over $\Q_{13}.$ Visible slopes: None

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q_{ 13 }$.

## Unramified/totally ramified tower

 Unramified subfield: $\Q_{13}$ Relative Eisenstein polynomial: $$x^{7} + 13$$ x^7 + 13

## Ramification polygon

 Residual polynomials: $z^{6} + 7z^{5} + 8z^{4} + 9z^{3} + 9z^{2} + 8z + 7$ Associated inertia: $2$ Indices of inseparability: $[0]$

## Invariants of the Galois closure

 Galois group: $D_7$ (as 7T2) Inertia group: $C_7$ (as 7T1) Wild inertia group: $C_1$ Unramified degree: $2$ Tame degree: $7$ Wild slopes: None Galois mean slope: $6/7$ Galois splitting model: $x^{7} - 3 x^{6} + 2 x^{5} + 6 x^{4} + 3 x^{3} - 16 x^{2} - 7 x - 7$