Base \(\Q_{2}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(11\)
Galois group $S_4\times C_2$ (as 6T11)

Related objects


Learn more

Defining polynomial

\(x^{6} + 4 x^{2} + 2\) Copy content Toggle raw display


Base field: $\Q_{2}$
Degree $d$: $6$
Ramification exponent $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $11$
Discriminant root field: $\Q_{2}(\sqrt{-2})$
Root number: $i$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[3]$

Intermediate fields

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}$
Relative Eisenstein polynomial: \( x^{6} + 4 x^{2} + 2 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 1$,$z^{4} + z^{2} + 1$
Associated inertia:$1$,$2$
Indices of inseparability:$[6, 0]$

Invariants of the Galois closure

Galois group:$C_2\times S_4$ (as 6T11)
Inertia group:$C_2\times A_4$ (as 6T6)
Wild inertia group:$C_2^3$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:$[4/3, 4/3, 3]$
Galois mean slope:$25/12$
Galois splitting model:$x^{6} + 2 x^{4} + 2$