Base \(\Q_{5}\)
Degree \(14\)
e \(2\)
f \(7\)
c \(7\)
Galois group $C_{14}$ (as 14T1)

Related objects

Learn more about

Defining polynomial

\(x^{14} - 250 x^{8} + 15625 x^{2} - 312500\)  Toggle raw display


Base field: $\Q_{5}$
Degree $d$: $14$
Ramification exponent $e$: $2$
Residue field degree $f$: $7$
Discriminant exponent $c$: $7$
Discriminant root field: $\Q_{5}(\sqrt{5})$
Root number: $1$
$|\Gal(K/\Q_{ 5 })|$: $14$
This field is Galois and abelian over $\Q_{5}.$

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: $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{7} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} - 5 t^{2} \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{14}$ (as 14T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$7$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:Not computed