Defining polynomial
| \( x^{12} + 4 x^{11} + 6 x^{10} + 4 x^{9} + 8 x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} - 4 x^{4} + 8 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $4$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $24$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $4$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.0.1, 2.6.6.1 |
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} + \left(8 t^{2} + 8 t + 4\right) x^{3} + \left(6 t^{2} - 2 t - 2\right) x^{2} + \left(-4 t^{2} - 4\right) x + 6 t^{2} - 6 t - 4 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_4^2:C_3$ (as 12T31) |
| Inertia group: | Intransitive group isomorphic to $C_4^2$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 3, 3] |
| Galois mean slope: | $21/8$ |
| Galois splitting model: | $x^{12} - 24 x^{10} + 152 x^{8} - 340 x^{6} + 335 x^{4} - 150 x^{2} + 25$ |