Properties

Label 2.6.4.1
Base \(\Q_{2}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(4\)
Galois group $S_3$ (as 6T2)

Related objects

Learn more about

Defining polynomial

\( x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9 \)

Invariants

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

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.3.2.1 x3

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}(\sqrt{*})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \)
Relative Eisenstein polynomial:$ x^{3} - 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$S_3$ (as 6T2)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$