Properties

Label 2.12.29.110
Base \(\Q_{2}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(29\)
Galois group 12T193

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Defining polynomial

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

Invariants

Base field: $\Q_{2}$
Degree $d$: $12$
Ramification exponent $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $29$
Discriminant root field: $\Q_{2}(\sqrt{-2\cdot 5})$
Root number: $-i$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[2,7/2]$

Intermediate fields

$\Q_{2}(\sqrt{-5})$, 2.3.2.1, 2.6.8.3

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}$
Relative Eisenstein polynomial: \( x^{12} + 8 x^{10} - 2 x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 6 \) Copy content Toggle raw display
Indices of inseparability:$[18, 6, 0]$

Invariants of the Galois closure

Galois group:12T193
Inertia group:$C_2^4:C_3.D_4$ (as 12T134)
Wild inertia group:$C_2^3.C_2^4$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:$[2, 8/3, 8/3, 3, 10/3, 10/3, 7/2]$
Galois mean slope:$10/3$
Galois splitting model: $x^{12} + 10 x^{10} + 32 x^{8} + 46 x^{6} + 66 x^{4} + 36 x^{2} + 54$ Copy content Toggle raw display