Properties

Label 3.9.12.23
Base \(\Q_{3}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(12\)
Galois group $C_3^2 : C_6$ (as 9T11)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{9} + 6 x^{4} + 3 x^{3} + 3\) Copy content Toggle raw display

Invariants

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

Intermediate fields

3.3.3.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}$
Relative Eisenstein polynomial: \( x^{9} + 6 x^{4} + 3 x^{3} + 3 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{4} + 2z + 1$
Associated inertia:$3$
Indices of inseparability:$[4, 3, 0]$

Invariants of the Galois closure

Galois group:$C_3^2:C_6$ (as 9T11)
Inertia group:$C_3:S_3$ (as 9T5)
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:$x^{9} - 3 x^{8} - 3 x^{7} + 15 x^{6} + 3 x^{5} - 51 x^{4} + 75 x^{3} - 54 x^{2} + 24$