Defining polynomial
| \( x^{12} + 8 x^{10} - 12 x^{8} - 24 x^{6} + 20 x^{4} - 16 x^{2} - 24 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $4$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $33$ |
| Discriminant root field: | $\Q_{2}(\sqrt{2*})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 2 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{2*})$, 2.3.0.1, 2.4.11.9, 2.6.9.3 |
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 + 16\right) x^{2} + 4 t^{2} + 2 t - 12 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{12}$ (as 12T1) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | [3, 4] |
| Galois mean slope: | $11/4$ |
| Galois splitting model: | $x^{12} - 4 x^{11} - 62 x^{10} + 224 x^{9} + 1199 x^{8} - 3812 x^{7} - 8598 x^{6} + 23764 x^{5} + 21370 x^{4} - 50444 x^{3} - 8360 x^{2} + 20312 x + 5041$ |