Properties

Label 7.6.4.3
Base \(\Q_{7}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(4\)
Galois group $C_6$ (as 6T1)

Related objects

Learn more

Defining polynomial

\(x^{6} + 56 x^{3} + 1323\)  Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $6$
Ramification exponent $e$: $3$
Residue field degree $f$: $2$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{7}(\sqrt{3})$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $6$
This field is Galois and abelian over $\Q_{7}.$

Intermediate fields

$\Q_{7}(\sqrt{3})$, 7.3.2.2

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

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Intransitive group isomorphic to $C_3$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{6} + 5 x^{4} + 6 x^{2} + 1$