Properties

Label 7.9.6.1
Base \(\Q_{7}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(6\)
Galois group $C_3^2$ (as 9T2)

Related objects

Learn more about

Defining polynomial

\( x^{9} + 42 x^{6} + 539 x^{3} + 2744 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $9$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{7}$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $9$
This field is Galois and abelian over $\Q_{7}$.

Intermediate fields

7.3.2.2, 7.3.2.1, 7.3.2.3, 7.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:7.3.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{3} - x + 2 \)
Relative Eisenstein polynomial:$ x^{3} - 7 t^{3} \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3^2$ (as 9T2)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$3$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{9} - 15 x^{7} - 4 x^{6} + 54 x^{5} + 12 x^{4} - 38 x^{3} - 9 x^{2} + 6 x + 1$