Defining polynomial
| \( x^{12} + 4 x^{11} - 2 x^{10} - 4 x^{9} + 8 x^{8} + 4 x^{6} + 8 x^{5} + 4 x^{2} + 8 x - 6 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $32$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.2.1, 2.6.10.4 |
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: | \( x^{12} - 12 x^{11} + 62 x^{10} - 180 x^{9} + 312 x^{8} - 288 x^{7} + 4 x^{6} + 360 x^{5} - 496 x^{4} + 352 x^{3} - 140 x^{2} + 24 x - 6 \) |
Invariants of the Galois closure
| Galois group: | $C_2\times C_4^2:C_3:C_2$ (as 12T97) |
| Inertia group: | 12T55 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [2, 8/3, 8/3, 11/3, 11/3] |
| Galois mean slope: | $10/3$ |
| Galois splitting model: | $x^{12} - 4 x^{10} + 5 x^{8} + 12 x^{6} - 25 x^{4} - 4 x^{2} + 49$ |