Properties

Label 3.9.9.7
Base \(\Q_{3}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(9\)
Galois group $C_3^2 : S_3 $ (as 9T13)

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Defining polynomial

\(x^{9} - 6 x^{7} + 9 x^{6} - 36 x^{5} - 36 x^{4} + 459 x^{3} - 108 x^{2} - 54 x + 27\) Copy content Toggle raw display

Invariants

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

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: \( x^{3} + \left(6 t^{2} + 6\right) x + 3 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_3^2:C_6$ (as 9T13)
Inertia group:Intransitive group isomorphic to $C_3:S_3$
Wild inertia group:$C_3^2$
Unramified degree:$3$
Tame degree:$2$
Wild slopes:$[3/2, 3/2]$
Galois mean slope:$25/18$
Galois splitting model:$x^{9} - x^{6} - 2 x^{3} + 1$