Properties

Label 29.14.13.1
Base \(\Q_{29}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(13\)
Galois group $C_{14}$ (as 14T1)

Related objects

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Defining polynomial

\(x^{14} - 29\)  Toggle raw display

Invariants

Base field: $\Q_{29}$
Degree $d$: $14$
Ramification exponent $e$: $14$
Residue field degree $f$: $1$
Discriminant exponent $c$: $13$
Discriminant root field: $\Q_{29}(\sqrt{29})$
Root number: $1$
$|\Gal(K/\Q_{ 29 })|$: $14$
This field is Galois and abelian over $\Q_{29}.$

Intermediate fields

$\Q_{29}(\sqrt{29})$, 29.7.6.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{29}$
Relative Eisenstein polynomial:\( x^{14} - 29 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{14}$ (as 14T1)
Inertia group:$C_{14}$
Unramified degree:$1$
Tame degree:$14$
Wild slopes:None
Galois mean slope:$13/14$
Galois splitting model:Not computed