# Properties

 Label 7.14.25.59 Base $$\Q_{7}$$ Degree $$14$$ e $$14$$ f $$1$$ c $$25$$ Galois group $C_{14}$ (as 14T1)

# Related objects

## Defining polynomial

 $$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$

## Invariants

 Base field: $\Q_{7}$ Degree $d$: $14$ Ramification exponent $e$: $14$ Residue field degree $f$: $1$ Discriminant exponent $c$: $25$ Discriminant root field: $\Q_{7}(\sqrt{7\cdot 3})$ Root number: $-i$ $|\Gal(K/\Q_{ 7 })|$: $14$ This field is Galois and abelian 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: $$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$

## Invariants of the Galois closure

 Galois group: $C_{14}$ (as 14T1) Inertia group: $C_{14}$ Unramified degree: $1$ Tame degree: $2$ Wild slopes: [2] Galois mean slope: $25/14$ Galois splitting model: $x^{14} - 28 x^{11} + 7 x^{10} + 14 x^{9} + 189 x^{8} - 90 x^{7} - 98 x^{6} - 196 x^{5} + 427 x^{4} - 217 x^{3} - 140 x^{2} + 119 x + 79$