Properties

Label 29.14.12.1
Base \(\Q_{29}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(12\)
Galois group $C_{14}$ (as 14T1)

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Defining polynomial

\(x^{14} + 2407 x^{7} + 1839267\)  Toggle raw display

Invariants

Base field: $\Q_{29}$
Degree $d$: $14$
Ramification exponent $e$: $7$
Residue field degree $f$: $2$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{29}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 29 })|$: $14$
This field is Galois and abelian over $\Q_{29}.$

Intermediate fields

$\Q_{29}(\sqrt{2})$, 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:$\Q_{29}(\sqrt{2})$ $\cong \Q_{29}(t)$ where $t$ is a root of \( x^{2} - x + 3 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{7} - 29 t^{7} \)$\ \in\Q_{29}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{14}$ (as 14T1)
Inertia group:Intransitive group isomorphic to $C_7$
Unramified degree:$2$
Tame degree:$7$
Wild slopes:None
Galois mean slope:$6/7$
Galois splitting model:Not computed