# Properties

 Label 7.14.14.19 Base $$\Q_{7}$$ Degree $$14$$ e $$7$$ f $$2$$ c $$14$$ Galois group $F_7 \times C_2$ (as 14T7)

# Learn more about

## Defining polynomial

 $$x^{14} + 35 x^{13} + 21 x^{12} + 28 x^{11} + 7 x^{10} + 7 x^{9} + 43 x^{7} + 28 x^{6} + 28 x^{5} + 21 x^{4} + 7 x^{2} + 42 x + 10$$

## Invariants

 Base field: $\Q_{7}$ Degree $d$ : $14$ Ramification exponent $e$ : $7$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $14$ Discriminant root field: $\Q_{7}(\sqrt{*})$ Root number: $1$ $|\Aut(K/\Q_{ 7 })|$: $2$ 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: $x^{7} + \left(14 t + 35\right) x^{6} + \left(14 t + 14\right) x^{5} + \left(21 t + 14\right) x^{4} + \left(14 t + 21\right) x^{3} + \left(21 t + 21\right) x^{2} + \left(28 t + 35\right) x + 7 t + 14 \in\Q_{7}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_2\times F_7$ (as 14T7) Inertia group: Intransitive group isomorphic to $F_7$ Unramified degree: $2$ Tame degree: $6$ Wild slopes: [7/6] Galois mean slope: $47/42$ Galois splitting model: $x^{14} - 14 x^{11} - 42 x^{10} - 84 x^{9} - 280 x^{8} - 1338 x^{7} - 4116 x^{6} - 11858 x^{5} - 25158 x^{4} - 39900 x^{3} - 50057 x^{2} - 46326 x - 17244$