Properties

Label 3.9.12.2
Base \(\Q_{3}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(12\)
Galois group $C_9$ (as 9T1)

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Defining polynomial

\(x^{9} + 18 x^{8} + 108 x^{7} + 243 x^{6} + 324 x^{5} + 972 x^{4} + 1269 x^{3} + 7614 x^{2} + 15984\) 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: $1$
$\card{ \Gal(K/\Q_{ 3 }) }$: $9$
This field is Galois and abelian 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: \( x^{3} + 6 x^{2} + 9 t^{2} + 18 t + 21 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_9$ (as 9T1)
Inertia group:Intransitive group isomorphic to $C_3$
Wild inertia group:$C_3$
Unramified degree:$3$
Tame degree:$1$
Wild slopes:$[2]$
Galois mean slope:$4/3$
Galois splitting model:$x^{9} - 57 x^{7} - 38 x^{6} + 855 x^{5} + 1254 x^{4} - 3192 x^{3} - 7524 x^{2} - 4275 x - 703$