Properties

Label 2.12.12.9
Base \(\Q_{2}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(12\)
Galois group $C_2^3:A_4$ (as 12T58)

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Defining polynomial

\(x^{12} + 2 x^{11} + 52 x^{10} + 116 x^{9} + 964 x^{8} + 2928 x^{7} + 10272 x^{6} + 24800 x^{5} + 47088 x^{4} + 67104 x^{3} + 107584 x^{2} + 64320 x + 107968\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $12$
Ramification exponent $e$: $2$
Residue field degree $f$: $6$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{2}$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[2]$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.3.0.1, 2.6.0.1

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

Unramified/totally ramified tower

Unramified subfield:2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{6} + x^{4} + x^{3} + x + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + \left(2 t^{5} + 2 t^{3} + 2 t^{2} + 2 t + 2\right) x + 4 t^{4} + 4 t^{3} + 4 t^{2} + 4 t + 6 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + t^{5} + t^{3} + t^{2} + t + 1$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois group:$C_2^3:A_4$ (as 12T58)
Inertia group:Intransitive group isomorphic to $C_2^4$
Wild inertia group:$C_2^4$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:$[2, 2, 2, 2]$
Galois mean slope:$15/8$
Galois splitting model: $x^{12} - 2 x^{10} - 7 x^{8} + 6 x^{6} + 5 x^{4} - 5 x^{2} + 1$ Copy content Toggle raw display