Properties

Label 3.9.12.18
Base \(\Q_{3}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(12\)
Galois group $(C_3^3:C_3):C_2$ (as 9T22)

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Defining polynomial

\(x^{9} - 12 x^{8} + 36 x^{7} + 90 x^{6} - 342 x^{5} + 432 x^{4} + 1971 x^{3} - 1890 x^{2} + 17037\) 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: $\Q_{3}(\sqrt{2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 3 }) }$: $1$
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: \( x^{3} + 3 t^{2} x^{2} + 18 t + 21 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_3^3:C_6$ (as 9T22)
Inertia group:Intransitive group isomorphic to $C_3^3$
Wild inertia group:$C_3^3$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:$[2, 2, 2]$
Galois mean slope:$52/27$
Galois splitting model:$x^{9} + 6 x^{7} - 8 x^{6} - 9 x^{5} + 3 x^{4} - 36 x^{3} - 18 x^{2} + 45 x + 29$