Defining polynomial
| \( x^{12} + 8 x^{11} + 4 x^{10} - 4 x^{9} + 8 x^{8} + 8 x^{7} - 4 x^{6} + 8 x^{5} - 4 x^{4} + 8 x^{3} + 8 x^{2} + 8 x - 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $32$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| 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, 2.4.10.2, 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: | \( x^{12} - 16 x^{11} + 124 x^{10} - 620 x^{9} + 2224 x^{8} - 6008 x^{7} + 12540 x^{6} - 20488 x^{5} + 26092 x^{4} - 25048 x^{3} + 16840 x^{2} - 6936 x + 1294 \) |
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: | $x^{12} - 4 x^{9} - 20 x^{8} - 48 x^{7} - 60 x^{6} - 64 x^{5} - 60 x^{4} - 48 x^{3} - 24 x^{2} - 8 x - 2$ |