Defining polynomial
| \( x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 8 x^{6} + 4 x^{4} - 4 x^{2} + 8 x + 6 \) |
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: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $4$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.2.1, 2.6.10.1 |
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: | \( x^{12} - 12 x^{11} + 66 x^{10} - 220 x^{9} + 496 x^{8} - 800 x^{7} + 952 x^{6} - 848 x^{5} + 564 x^{4} - 272 x^{3} + 92 x^{2} - 24 x + 6 \) |
Invariants of the Galois closure
| Galois group: | $C_2\times C_4^2:C_3:C_2$ (as 12T95) |
| Inertia group: | 12T55 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [8/3, 8/3, 3, 11/3, 11/3] |
| Galois mean slope: | $163/48$ |
| Galois splitting model: | $x^{12} - 4 x^{10} + 7 x^{8} - 16 x^{6} + 17 x^{4} - 8 x^{2} + 1$ |