Defining polynomial
\(x^{12} + 8 x^{10} - 2 x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 6\)
|
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: |
\( x^{12} + 8 x^{10} - 2 x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 6 \)
|
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: |
$x^{12} + 10 x^{10} + 32 x^{8} + 46 x^{6} + 66 x^{4} + 36 x^{2} + 54$
|