# Properties

 Label 7.14.15.5 Base $$\Q_{7}$$ Degree $$14$$ e $$14$$ f $$1$$ c $$15$$ Galois group $F_7$ (as 14T4)

# Related objects

## Defining polynomial

 $$x^{14} - 21 x^{13} + 7 x^{12} + 14 x^{11} - 21 x^{9} + 7 x^{7} + 14 x^{6} - 21 x^{4} + 14 x^{3} + 7 x^{2} + 14$$

## Invariants

 Base field: $\Q_{7}$ Degree $d$ : $14$ Ramification exponent $e$ : $14$ Residue field degree $f$ : $1$ Discriminant exponent $c$ : $15$ Discriminant root field: $\Q_{7}(\sqrt{7*})$ Root number: $-i$ $|\Aut(K/\Q_{ 7 })|$: $2$ 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: $\Q_{7}$ Relative Eisenstein polynomial: $$x^{14} - 21 x^{13} + 7 x^{12} + 14 x^{11} - 21 x^{9} + 7 x^{7} + 14 x^{6} - 21 x^{4} + 14 x^{3} + 7 x^{2} + 14$$

## Invariants of the Galois closure

 Galois group: $F_7$ (as 14T4) Inertia group: $F_7$ Unramified degree: $1$ Tame degree: $6$ Wild slopes: [7/6] Galois mean slope: $47/42$ Galois splitting model: $x^{14} - 7 x^{13} + 28 x^{12} - 63 x^{11} + 98 x^{10} - 133 x^{9} + 147 x^{8} - 53 x^{7} - 91 x^{6} - 35 x^{5} + 469 x^{4} - 707 x^{3} + 497 x^{2} - 175 x + 25$