Defining polynomial
| \( x^{12} - 4 x^{11} + 4 x^{10} + 8 x^{9} + 6 x^{8} + 8 x^{7} - 6 x^{6} + 4 x^{5} + 2 x^{2} + 4 x + 6 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $24$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $4$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.2.1, 2.6.6.8, 2.6.11.4, 2.6.11.7 |
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: | \( x^{12} - 4 x^{11} + 4 x^{10} + 8 x^{9} + 6 x^{8} + 8 x^{7} - 6 x^{6} + 4 x^{5} + 2 x^{2} + 4 x + 6 \) |
Invariants of the Galois closure
| Galois group: | $C_2\times C_2^2:S_4$ (as 12T103) |
| Inertia group: | 12T56 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3, 8/3, 8/3, 3] |
| Galois mean slope: | $127/48$ |
| Galois splitting model: | $x^{12} + 2 x^{10} + 5 x^{8} + 6 x^{6} + 5 x^{4} + 2 x^{2} + 1$ |