# Properties

 Label 13.7.0.1 Base $$\Q_{13}$$ Degree $$7$$ e $$1$$ f $$7$$ c $$0$$ Galois group $C_7$ (as 7T1)

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## Defining polynomial

 $$x^{7} + 3 x + 11$$ x^7 + 3*x + 11

## Invariants

 Base field: $\Q_{13}$ Degree $d$: $7$ Ramification exponent $e$: $1$ Residue field degree $f$: $7$ Discriminant exponent $c$: $0$ Discriminant root field: $\Q_{13}$ Root number: $1$ $\card{ \Gal(K/\Q_{ 13 }) }$: $7$ This field is Galois and abelian 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: 13.7.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of $$x^{7} + 3 x + 11$$ x^7 + 3*x + 11 Relative Eisenstein polynomial: $$x - 13$$ x - 13 $\ \in\Q_{13}(t)[x]$

## Ramification polygon

The ramification polygon is trivial for unramified extensions.

## Invariants of the Galois closure

 Galois group: $C_7$ (as 7T1) Inertia group: trivial Wild inertia group: $C_1$ Unramified degree: $7$ Tame degree: $1$ Wild slopes: None Galois mean slope: $0$ Galois splitting model: $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$