Defining polynomial
| \( x^{12} + 4 x^{11} + 8 x^{10} - 4 x^{9} + 8 x^{8} + 8 x^{6} + 8 x^{4} + 4 x^{2} + 8 x - 6 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $32$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.2.1, 2.6.11.15 |
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: | \( x^{12} - 12 x^{11} + 56 x^{10} - 132 x^{9} + 4200 x^{8} - 19568 x^{7} + 133736 x^{6} - 1774096 x^{5} + 2903000 x^{4} - 21859008 x^{3} + 166229428 x^{2} - 332005768 x + 206385162 \) |
Invariants of the Galois closure
| Galois group: | 12T250 |
| Inertia group: | $D_4\wr C_3$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3, 2, 8/3, 8/3, 3, 10/3, 10/3, 7/2] |
| Galois mean slope: | $2563/768$ |
| Galois splitting model: | $x^{12} - 12 x^{10} + 36 x^{8} + 4740 x^{6} + 17820 x^{4} - 5400 x^{2} - 900$ |