Properties

Label 2.12.24.36
Base \(\Q_{2}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(24\)
Galois group $C_2\wr C_6$ (as 12T134)

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Defining polynomial

\(x^{12} + 12 x^{11} + 66 x^{10} + 212 x^{9} + 470 x^{8} + 896 x^{7} + 1712 x^{6} + 2944 x^{5} + 3860 x^{4} + 4592 x^{3} + 5512 x^{2} + 3984 x + 3576\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $12$
Ramification exponent $e$: $4$
Residue field degree $f$: $3$
Discriminant exponent $c$: $24$
Discriminant root field: $\Q_{2}(\sqrt{5})$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[2, 3]$

Intermediate fields

$\Q_{2}(\sqrt{-5})$, 2.3.0.1, 2.6.6.5

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: \( x^{4} + 4 x^{3} + \left(4 t + 6\right) x^{2} + \left(4 t^{2} + 4 t + 4\right) x + 8 t^{2} + 8 t + 14 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_2\wr C_6$ (as 12T134)
Inertia group:Intransitive group isomorphic to $C_2^3:D_4$
Wild inertia group:$C_2^3:D_4$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:$[2, 2, 2, 2, 3, 3]$
Galois mean slope:$87/32$
Galois splitting model:$x^{12} - 6 x^{10} + 3 x^{8} + 24 x^{6} - 36 x^{4} + 18 x^{2} - 3$