Properties

Label 29.9.6.1
Base \(\Q_{29}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(6\)
Galois group $S_3\times C_3$ (as 9T4)

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Defining polynomial

\(x^{9} + 6 x^{7} + 168 x^{6} + 12 x^{5} + 150 x^{4} - 11725 x^{3} + 672 x^{2} - 10416 x + 175848\) Copy content Toggle raw display

Invariants

Base field: $\Q_{29}$
Degree $d$: $9$
Ramification exponent $e$: $3$
Residue field degree $f$: $3$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{29}(\sqrt{2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 29 }) }$: $3$
This field is not Galois over $\Q_{29}.$
Visible slopes:None

Intermediate fields

29.3.2.1, 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:29.3.0.1 $\cong \Q_{29}(t)$ where $t$ is a root of \( x^{3} + 2 x + 27 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + 29 \) $\ \in\Q_{29}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{2} + 3z + 3$
Associated inertia:$2$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_3\times S_3$ (as 9T4)
Inertia group:Intransitive group isomorphic to $C_3$
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:Not computed