Properties

Label 2.6.9.2
Base \(\Q_{2}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(9\)
Galois group $A_4\times C_2$ (as 6T6)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{6} + 4 x^{5} - 10 x^{4} + 160 x^{3} + 1212 x^{2} + 2160 x - 1048\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $6$
Ramification exponent $e$: $2$
Residue field degree $f$: $3$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{2}(\sqrt{2})$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[3]$

Intermediate fields

2.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} + x + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + \left(4 t^{2} + 4\right) x + 8 t^{2} + 12 t + 2 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 1$
Associated inertia:$1$
Indices of inseparability:$[2, 0]$

Invariants of the Galois closure

Galois group:$C_2\times A_4$ (as 6T6)
Inertia group:Intransitive group isomorphic to $C_2^3$
Wild inertia group:$C_2^3$
Unramified degree:$3$
Tame degree:$1$
Wild slopes:$[2, 2, 3]$
Galois mean slope:$9/4$
Galois splitting model:$x^{6} - 12 x^{2} - 8$