Defining polynomial
| \( x^{12} + 4 x^{11} + 8 x^{10} + 4 x^{9} - 2 x^{8} + 8 x^{7} + 4 x^{6} + 6 x^{4} + 8 x - 2 \) |
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
| 2.3.2.1, 2.6.11.4 |
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: | \( x^{12} - 12 x^{11} + 136 x^{10} - 1084 x^{9} - 4882 x^{8} + 115768 x^{7} - 1562508 x^{6} + 11368080 x^{5} - 71618202 x^{4} + 389374384 x^{3} - 1210962096 x^{2} + 1776256968 x - 945196274 \) |
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: | $x^{12} - 6 x^{10} - 12 x^{9} + 5 x^{8} + 48 x^{7} + 68 x^{6} + 8 x^{5} - 107 x^{4} - 176 x^{3} - 146 x^{2} - 68 x - 13$ |