Properties

Label 2.6.10.2
Base \(\Q_{2}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(10\)
Galois group $S_4$ (as 6T7)

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Defining polynomial

\(x^{6} + 2 x^{5} + 4 x^{4} + 4 x + 2\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $6$
Ramification exponent $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{2}$
Root number: $1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[8/3]$

Intermediate fields

2.3.2.1

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: \( x^{6} + 2 x^{5} + 4 x^{4} + 4 x + 2 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 1$,$z^{4} + z^{2} + 1$
Associated inertia:$1$,$2$
Indices of inseparability:$[5, 0]$

Invariants of the Galois closure

Galois group:$S_4$ (as 6T7)
Inertia group:$A_4$ (as 6T4)
Wild inertia group:$C_2^2$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:$[8/3, 8/3]$
Galois mean slope:$13/6$
Galois splitting model:$x^{6} + 3 x^{4} + 3 x^{2} - 1$