Defining polynomial
| \( x^{12} + 12 x^{11} - 8 x^{9} + 12 x^{8} - 8 x^{7} + 4 x^{6} + 8 x^{5} - 12 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 8 \) |
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: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.3.0.1, 2.6.6.3 |
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(4 t^{2} + 4 t + 4\right) x^{3} - 2 t x^{2} + 4 t x - 2 t^{2} + 4 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_2^2\wr C_2:C_3$ (as 12T87) |
| Inertia group: | Intransitive group isomorphic to $C_2^2\wr C_2$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 2, 3, 3] |
| Galois mean slope: | $43/16$ |
| Galois splitting model: | $x^{12} + 10 x^{10} - 137 x^{8} + 176 x^{6} + 511 x^{4} + 294 x^{2} + 49$ |