Defining polynomial
| \( x^{12} + 8 x^{11} - 6 x^{10} - 6 x^{8} + 8 x^{7} + 8 x^{6} - 4 x^{5} + 8 x^{3} + 8 x^{2} + 8 x - 6 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $28$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $4$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.2.1, 2.6.10.6, 2.6.11.6, 2.6.11.16 |
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: | \( x^{12} + 8 x^{11} - 6 x^{10} - 6 x^{8} + 8 x^{7} + 8 x^{6} - 4 x^{5} + 8 x^{3} + 8 x^{2} + 8 x - 6 \) |
Invariants of the Galois closure
| Galois group: | $C_2^2\times C_2^2:S_4$ (as 12T139) |
| Inertia group: | 12T90 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3, 2, 8/3, 8/3, 3] |
| Galois mean slope: | $259/96$ |
| Galois splitting model: | $x^{12} - 4 x^{11} + 24 x^{10} - 64 x^{9} + 182 x^{8} - 360 x^{7} + 640 x^{6} - 928 x^{5} + 1052 x^{4} - 1024 x^{3} + 752 x^{2} - 320 x + 56$ |