Properties

Label 5.14.7.2
Base \(\Q_{5}\)
Degree \(14\)
e \(2\)
f \(7\)
c \(7\)
Galois group $C_{14}$ (as 14T1)

Related objects

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Defining polynomial

\( x^{14} - 15625 x^{2} + 156250 \)

Invariants

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

$\Q_{5}(\sqrt{5*})$, 5.7.0.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:5.7.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{7} - x + 2 \)
Relative Eisenstein polynomial:$ x^{2} - 5 t \in\Q_{5}(t)[x]$

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