Defining polynomial
| \( x^{12} + 8 x^{10} + 4 x^{9} + 8 x^{8} + 8 x^{6} + 4 x^{4} + 8 x^{3} + 8 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $4$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $18$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.0.1, 2.6.6.4 |
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(-2 t^{2} - 2 t + 4\right) x^{3} + \left(4 t^{2} + 4 t + 4\right) x^{2} + \left(4 t^{2} + 4 t + 4\right) x - 2 t^{2} + 2 t + 4 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2^5.(C_2\times C_6)$ (as 12T142) |
| Inertia group: | Intransitive group isomorphic to $C_2^6$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 2, 2, 2, 2] |
| Galois mean slope: | $63/32$ |
| Galois splitting model: | $x^{12} - 2 x^{11} - 18 x^{10} + 64 x^{9} + 12 x^{8} - 362 x^{7} + 642 x^{6} - 338 x^{5} - 221 x^{4} + 312 x^{3} - 52 x^{2} - 52 x + 13$ |