Properties

Label 2.6.6.5
Base \(\Q_{2}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(6\)
Galois group $C_6$ (as 6T1)

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Defining polynomial

\(x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $6$
Ramification exponent $e$: $2$
Residue field degree $f$: $3$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{2}(\sqrt{-5})$
Root number: $-i$
$\card{ \Gal(K/\Q_{ 2 }) }$: $6$
This field is Galois and abelian over $\Q_{2}.$
Visible slopes:$[2]$

Intermediate fields

$\Q_{2}(\sqrt{-5})$, 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: \( x^{2} + 2 x + 6 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_2$
Unramified degree:$3$
Tame degree:$1$
Wild slopes:$[2]$
Galois mean slope:$1$
Galois splitting model:$x^{6} - 6 x^{4} + 9 x^{2} - 3$