Defining polynomial
| \( x^{12} - 36 x^{10} + 972 x^{8} + 1696 x^{6} - 1168 x^{4} + 1984 x^{2} + 1344 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $2$ |
| Residue field degree $f$ : | $6$ |
| 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, 2.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{6} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{2} + 8 t^{5} + 8 t^{4} + 4 t^{3} + 4 t^{2} + 8 t - 2 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2^5.C_6$ (as 12T105) |
| Inertia group: | Intransitive group isomorphic to $C_2^5$ |
| Unramified degree: | $6$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 2, 2, 3] |
| Galois mean slope: | $39/16$ |
| Galois splitting model: | $x^{12} - 52 x^{8} + 104 x^{6} + 416 x^{4} - 1248 x^{2} + 832$ |