Properties

Label 2.12.33.421
Base \(\Q_{2}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(33\)
Galois group $C_4^2:C_3:C_2^2$ (as 12T115)

Related objects

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Defining polynomial

\( x^{12} + 2 x^{10} + 2 x^{8} + 4 x^{6} - 2 x^{4} + 8 x^{2} + 6 \)

Invariants

Base field: $\Q_{2}$
Degree $d$: $12$
Ramification exponent $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $33$
Discriminant root field: $\Q_{2}(\sqrt{-2*})$
Root number: $-i$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}.$

Intermediate fields

2.3.2.1, 2.6.10.1

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:\( x^{12} + 2 x^{10} + 2 x^{8} + 4 x^{6} - 2 x^{4} + 8 x^{2} + 6 \)

Invariants of the Galois closure

Galois group:$C_4^2:C_3:C_2^2$ (as 12T115)
Inertia group:12T61
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[8/3, 8/3, 3, 23/6, 23/6]
Galois mean slope:$169/48$
Galois splitting model:$x^{12} + 12 x^{10} + 22 x^{8} + 1452 x^{4} + 7744 x^{2} + 10648$