Defining polynomial
| \( x^{12} + 14 x^{11} + 12 x^{10} + 4 x^{9} + 10 x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 16 x - 8 \) |
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
| $\Q_{2}(\sqrt{-*})$, 2.3.0.1, 2.6.6.5 |
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} + 4 t\right) x^{3} + \left(2 t^{2} + 4 t + 2\right) x^{2} + 4 t^{2} x - 2 t^{2} - 2 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $D_4\times A_4$ (as 12T51) |
| Inertia group: | Intransitive group isomorphic to $C_2^4$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 2, 2] |
| Galois mean slope: | $15/8$ |
| Galois splitting model: | $x^{12} - 6 x^{10} - 2 x^{9} + 21 x^{8} + 6 x^{7} - 42 x^{6} - 12 x^{5} + 48 x^{4} + 10 x^{3} - 24 x^{2} + 1$ |