Defining polynomial
| \( x^{12} + 8 x^{11} - 4 x^{10} + 4 x^{9} + 4 x^{8} + 8 x^{7} - 4 x^{6} + 8 x^{4} + 8 x^{2} + 8 x + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $32$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-*})$ |
| 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, 2.6.11.9 |
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: | \( x^{12} - 16 x^{11} + 132 x^{10} - 676 x^{9} + 2092 x^{8} - 1912 x^{7} - 17332 x^{6} + 104704 x^{5} - 310616 x^{4} + 557120 x^{3} - 593496 x^{2} + 320280 x - 66350 \) |
Invariants of the Galois closure
| Galois group: | $D_4\times S_4$ (as 12T86) |
| Inertia group: | 12T51 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [2, 8/3, 8/3, 3, 7/2] |
| Galois mean slope: | $37/12$ |
| Galois splitting model: | $x^{12} + 2 x^{10} + 13 x^{8} + 12 x^{6} + 23 x^{4} + 26 x^{2} + 11$ |