Properties

Label 2.12.24.424
Base \(\Q_{2}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(24\)
Galois group $C_4^2:C_3.D_4$ (as 12T149)

Related objects

Learn more

Defining polynomial

\(x^{12} + 2 x^{10} + 4 x^{7} - 2 x^{6} + 4 x^{4} + 4 x^{3} - 2 x^{2} + 4 x + 2\)  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: $\Q_{2}(\sqrt{-1})$
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:\( x^{12} + 2 x^{10} + 4 x^{7} - 2 x^{6} + 4 x^{4} + 4 x^{3} - 2 x^{2} + 4 x + 2 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_4^2:C_3.D_4$ (as 12T149)
Inertia group:12T92
Wild inertia group:$C_2\times C_4:D_4$
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:$x^{12} + 6 x^{10} + 25 x^{8} + 132 x^{6} + 495 x^{4} + 1100 x^{2} + 1375$  Toggle raw display