Properties

Label 2.12.18.15
Base \(\Q_{2}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(18\)
Galois group $C_6\times C_2$ (as 12T2)

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Defining polynomial

\(x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $12$
Ramification exponent $e$: $2$
Residue field degree $f$: $6$
Discriminant exponent $c$: $18$
Discriminant root field: $\Q_{2}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 2 }) }$: $12$
This field is Galois and abelian over $\Q_{2}.$
Visible slopes:$[3]$

Intermediate fields

$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.3.0.1, 2.4.6.2, 2.6.0.1, 2.6.9.5, 2.6.9.7

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

Unramified/totally ramified tower

Unramified subfield:2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{6} + x^{4} + x^{3} + x + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + \left(4 t^{5} + 4 t^{4} + 4 t^{3} + 4 t^{2}\right) x + 8 t^{5} + 4 t^{3} + 4 t^{2} + 8 t + 14 \) $\ \in\Q_{2}(t)[x]$ 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:$C_2\times C_6$ (as 12T2)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_2$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:$[3]$
Galois mean slope:$3/2$
Galois splitting model:$x^{12} - 8 x^{6} + 64$