Properties

Label 2.12.25.120
Base \(\Q_{2}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(25\)
Galois group 12T224

Related objects

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Defining polynomial

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

Invariants

Base field: $\Q_{2}$
Degree $d$: $12$
Ramification exponent $e$: $12$
Residue field degree $f$: $1$
Discriminant exponent $c$: $25$
Discriminant root field: $\Q_{2}(\sqrt{-2\cdot 5})$
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:\( x^{12} + 8 x^{10} - 2 x^{8} + 8 x^{6} + 8 x^{4} - 6 x^{2} + 6 \)  Toggle raw display

Invariants of the Galois closure

Galois group:12T224
Inertia group:12T188
Wild inertia group:$C_2^6.C_2^2$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[4/3, 4/3, 8/3, 8/3, 3, 3, 19/6, 19/6]
Galois mean slope:$1183/384$
Galois splitting model:$x^{12} - 12 x^{10} + 62 x^{8} - 164 x^{6} + 228 x^{4} - 162 x^{2} + 54$  Toggle raw display