Defining polynomial
| \( x^{12} + 4 x^{11} + 4 x^{10} + 4 x^{8} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 8 x^{2} - 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $34$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| 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: | \( x^{12} - 12 x^{11} + 68 x^{10} - 240 x^{9} + 580 x^{8} - 992 x^{7} + 1204 x^{6} - 1016 x^{5} + 562 x^{4} - 168 x^{3} - 8 x^{2} + 32 x - 18 \) |
Invariants of the Galois closure
| Galois group: | $C_2\times C_4^2:C_3:C_2^2$ (as 12T138) |
| Inertia group: | 12T89 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [2, 8/3, 8/3, 3, 23/6, 23/6] |
| Galois mean slope: | $85/24$ |
| Galois splitting model: | $x^{12} + 2 x^{10} + 59 x^{8} - 120 x^{6} + 69 x^{4} - 14 x^{2} + 1$ |