Properties

Label 29.6.0.1
Base \(\Q_{29}\)
Degree \(6\)
e \(1\)
f \(6\)
c \(0\)
Galois group $C_6$ (as 6T1)

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Defining polynomial

\(x^{6} + x^{4} + 25 x^{3} + 17 x^{2} + 13 x + 2\) Copy content Toggle raw display

Invariants

Base field: $\Q_{29}$
Degree $d$: $6$
Ramification exponent $e$: $1$
Residue field degree $f$: $6$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{29}(\sqrt{2})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 29 }) }$: $6$
This field is Galois and abelian over $\Q_{29}.$
Visible slopes:None

Intermediate fields

$\Q_{29}(\sqrt{2})$, 29.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:29.6.0.1 $\cong \Q_{29}(t)$ where $t$ is a root of \( x^{6} + x^{4} + 25 x^{3} + 17 x^{2} + 13 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 29 \) $\ \in\Q_{29}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{6} - x^{5} + 2 x^{4} + 8 x^{3} - x^{2} - 5 x + 7$