Properties

Label 3.9.12.8
Base \(\Q_{3}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(12\)
Galois group $C_3 \wr C_3 $ (as 9T17)

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Defining polynomial

\(x^{9} - 24 x^{8} + 324 x^{7} - 405 x^{6} - 36 x^{5} + 4860 x^{4} + 540 x^{3} + 6912 x^{2} + 7236\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $9$
Ramification exponent $e$: $3$
Residue field degree $f$: $3$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{3}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 3 }) }$: $3$
This field is not Galois over $\Q_{3}.$
Visible slopes:$[2]$

Intermediate fields

3.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:3.3.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{3} + 2 x + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + \left(6 t^{2} + 6 t\right) x^{2} + 9 t^{2} + 9 t + 21 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{2} + 2t^{2} + 2t$
Associated inertia:$1$
Indices of inseparability:$[2, 0]$

Invariants of the Galois closure

Galois group:$C_3\wr C_3$ (as 9T17)
Inertia group:Intransitive group isomorphic to $C_3^3$
Wild inertia group:$C_3^3$
Unramified degree:$3$
Tame degree:$1$
Wild slopes:$[2, 2, 2]$
Galois mean slope:$52/27$
Galois splitting model:$x^{9} - 87 x^{7} - 144 x^{6} + 1926 x^{5} + 5367 x^{4} - 2950 x^{3} - 11853 x^{2} + 2532 x + 3833$