Defining polynomial
| \( x^{12} + 32 x^{11} - 10 x^{10} + 8 x^{9} - 18 x^{8} + 32 x^{7} + 20 x^{6} + 24 x^{5} - 24 x^{4} + 32 x^{3} + 16 x^{2} - 24 \) |
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: | $-1$ |
| $|\Gal(K/\Q_{ 2 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{2*})$, $\Q_{2}(\sqrt{-2*})$, 2.3.0.1, 2.4.8.1, 2.6.6.3, 2.6.9.3, 2.6.9.7 |
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} + 8 t + 8\right) x^{3} + \left(-2 t^{2} - 2\right) x^{2} + \left(-4 t^{2} + 8 t\right) x + 6 t^{2} + 6 t - 2 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_6$ (as 12T2) |
| Inertia group: | Intransitive group isomorphic to $C_2^2$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 3] |
| Galois mean slope: | $2$ |
| Galois splitting model: | $x^{12} + 18 x^{8} + 45 x^{4} + 9$ |