Defining polynomial
| \( x^{12} - 4 x^{11} - 4 x^{10} + 4 x^{9} - 6 x^{8} + 8 x^{6} + 8 x^{5} + 8 x^{4} + 8 x + 2 \) |
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
| $\Q_{2}(\sqrt{2})$, 2.3.2.1, 2.4.10.3, 2.6.11.1 |
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: | \( x^{12} - 4 x^{11} - 4 x^{10} + 84 x^{9} - 294 x^{8} - 1296 x^{7} + 3912 x^{6} + 8376 x^{5} - 15688 x^{4} + 5664 x^{3} + 15552 x^{2} - 140616 x + 32242 \) |
Invariants of the Galois closure
| Galois group: | $S_3\times D_4$ (as 12T28) |
| Inertia group: | $D_4 \times C_3$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [2, 3, 7/2] |
| Galois mean slope: | $17/6$ |
| Galois splitting model: | $x^{12} + 4 x^{10} - 28 x^{9} + 24 x^{8} - 80 x^{7} + 292 x^{6} - 416 x^{5} + 548 x^{4} - 848 x^{3} + 704 x^{2} - 232 x + 14$ |