Defining polynomial
| \( x^{12} - 2 x^{10} - 5 x^{8} - 12 x^{6} - 13 x^{4} + 6 x^{2} + 1 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $4$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $30$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.0.1, 2.6.9.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{4} - 4 x^{3} + 8 t^{2} x^{2} + 8 x + 2 t^{2} - 4 t + 4 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_4:D_4:C_3$ (as 12T60) |
| Inertia group: | Intransitive group isomorphic to $C_4:D_4$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 3, 7/2, 7/2] |
| Galois mean slope: | $51/16$ |
| Galois splitting model: | $x^{12} + 4 x^{10} - 82 x^{8} + 330 x^{6} - 549 x^{4} + 338 x^{2} + 1$ |