Properties

Label 2.12.16.1
Base \(\Q_{2}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(16\)
Galois group $A_4^2:D_4$ (as 12T208)

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Defining polynomial

\(x^{12} - 2 x^{11} + 4 x^{10} + 4 x^{9} - 4 x^{7} + 10 x^{6} + 8 x^{5} + 4 x^{4} + 12 x^{3} + 12 x^{2} + 12\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $12$
Ramification exponent $e$: $6$
Residue field degree $f$: $2$
Discriminant exponent $c$: $16$
Discriminant root field: $\Q_{2}(\sqrt{-5})$
Root number: $i$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[2]$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.6.4.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} + x + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{6} + 2 t x^{5} + 2 x^{3} + 2 x^{2} + 2 t + 4 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + t + 1$,$z^{4} + z^{2} + 1$
Associated inertia:$1$,$1$
Indices of inseparability:$[3, 0]$

Invariants of the Galois closure

Galois group:$A_4^2:D_4$ (as 12T208)
Inertia group:Intransitive group isomorphic to $C_2^2\wr C_3$
Wild inertia group:$C_2^6$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:$[4/3, 4/3, 4/3, 4/3, 2, 2]$
Galois mean slope:$175/96$
Galois splitting model: $x^{12} - 12 x^{10} - 44 x^{9} + 24 x^{8} + 276 x^{7} + 558 x^{6} + 192 x^{5} - 1860 x^{4} - 3356 x^{3} - 1776 x^{2} - 168 x - 4$ Copy content Toggle raw display