Defining polynomial
| \( x^{12} + 4 x^{11} - 4 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 6 x^{4} + 8 x^{2} + 8 x + 6 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $32$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-*})$ |
| 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: | \( x^{12} - 4 x^{11} - 192 x^{10} - 8076 x^{9} + 72226 x^{8} + 1366136 x^{7} + 17984952 x^{6} - 242555104 x^{5} - 3154451994 x^{4} + 9114999664 x^{3} + 121486031464 x^{2} + 284428165560 x + 297704982742 \) |
Invariants of the Galois closure
| Galois group: | 12T186 |
| Inertia group: | 12T142 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [2, 8/3, 8/3, 3, 10/3, 10/3, 7/2] |
| Galois mean slope: | $10/3$ |
| Galois splitting model: | $x^{12} - 2 x^{10} + 13 x^{8} - 100 x^{6} + 135 x^{4} - 66 x^{2} + 11$ |