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

Learn more about

Defining polynomial

\( x^{9} + 6 x^{4} + 3 x^{3} + 3 \)

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: $1$
$|\Aut(K/\Q_{ 3 })|$: $1$
This field is not Galois over $\Q_{3}$.

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:\( x^{9} + 6 x^{4} + 3 x^{3} + 3 \)

Invariants of the Galois closure

Galois group:$He_3:C_2$ (as 9T11)
Inertia group:$C_3^2:C_2$
Unramified degree:$3$
Tame degree:$2$
Wild slopes:[3/2, 3/2]
Galois mean slope:$25/18$
Galois splitting model:$x^{9} - 3 x^{8} - 3 x^{7} + 15 x^{6} + 3 x^{5} - 51 x^{4} + 75 x^{3} - 54 x^{2} + 24$