Defining polynomial
| \( x^{12} + 36 x^{11} - 48 x^{10} - 12 x^{9} - 58 x^{8} - 32 x^{6} + 48 x^{5} + 60 x^{4} - 16 x^{3} + 64 x^{2} - 48 x + 40 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $4$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $24$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $1$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{4} + 4 t^{2} x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(4 t + 4\right) x - 2 t^{2} + 2 t + 4 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | 12T205 |
| Inertia group: | Intransitive group isomorphic to $C_2^6:C_3$ |
| Unramified degree: | $6$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3, 4/3, 4/3, 8/3, 8/3] |
| Galois mean slope: | $223/96$ |
| Galois splitting model: | $x^{12} + 1176 x^{10} - 9352 x^{9} + 13158 x^{8} + 2285928 x^{7} - 15228264 x^{6} + 14779224 x^{5} + 1569752496 x^{4} - 1444920928 x^{3} + 3813990768 x^{2} - 18616281792 x + 23067540424$ |