# Properties

 Label 7.14.16.1 Base $$\Q_{7}$$ Degree $$14$$ e $$7$$ f $$2$$ c $$16$$ Galois group $C_7^2:C_6$ (as 14T14)

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## Defining polynomial

 $$x^{14} + 28 x^{13} + 21 x^{12} + 42 x^{11} + 14 x^{10} + 28 x^{8} + 37 x^{7} + 28 x^{5} + 7 x^{4} + 7 x^{3} + 35 x^{2} + 28 x + 31$$

## Invariants

 Base field: $\Q_{7}$ Degree $d$ : $14$ Ramification exponent $e$ : $7$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $16$ Discriminant root field: $\Q_{7}(\sqrt{*})$ Root number: $1$ $|\Aut(K/\Q_{ 7 })|$: $1$ This field is not Galois over $\Q_{7}$.

## 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: $\Q_{7}(\sqrt{*})$ $\cong \Q_{7}(t)$ where $t$ is a root of $$x^{2} - x + 3$$ Relative Eisenstein polynomial: $x^{7} + 7 t x^{6} + \left(21 t + 42\right) x^{5} + \left(21 t + 28\right) x^{4} + \left(21 t + 35\right) x^{3} + \left(7 t + 21\right) x^{2} + 42 t + 21 \in\Q_{7}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_7^2:C_6$ (as 14T14) Inertia group: Intransitive group isomorphic to $C_7^2:C_3$ Unramified degree: $2$ Tame degree: $3$ Wild slopes: [4/3, 4/3] Galois mean slope: $194/147$ Galois splitting model: $x^{14} - 84 x^{12} - 84 x^{11} + 3206 x^{10} + 4956 x^{9} - 66668 x^{8} - 130154 x^{7} + 786842 x^{6} + 1801856 x^{5} - 4775288 x^{4} - 13285916 x^{3} + 9985696 x^{2} + 39242252 x + 21880010$