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

Learn more about

Defining polynomial

\(x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126\)  Toggle raw display

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

$\Q_{7}(\sqrt{7\cdot 3})$, 7.7.12.1

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:\( x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126 \)  Toggle raw display

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:$x^{14} - 28 x^{11} + 7 x^{10} + 14 x^{9} + 189 x^{8} - 90 x^{7} - 98 x^{6} - 196 x^{5} + 427 x^{4} - 217 x^{3} - 140 x^{2} + 119 x + 79$