Properties

Label 2.12.24.377
Base \(\Q_{2}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(24\)
Galois group $C_2^4:C_3.D_4$ (as 12T146)

Related objects

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Defining polynomial

\(x^{12} - 4 x^{11} + 2 x^{8} - 4 x^{6} + 8 x^{3} - 2 x^{2} - 4 x + 6\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $12$
Ramification exponent $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $24$
Discriminant root field: $\Q_{2}(\sqrt{-5})$
Root number: $i$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}.$

Intermediate fields

2.3.2.1, 2.6.6.8

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:\( x^{12} - 4 x^{11} + 2 x^{8} - 4 x^{6} + 8 x^{3} - 2 x^{2} - 4 x + 6 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2^4:C_3.D_4$ (as 12T146)
Inertia group:12T88
Wild inertia group:$C_2\times C_2^2\wr C_2$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[4/3, 4/3, 2, 7/3, 7/3, 3]
Galois mean slope:$247/96$
Galois splitting model:$x^{12} - 18 x^{10} + 72 x^{8} + 192 x^{6} + 72 x^{4} - 72 x^{2} + 12$  Toggle raw display