Defining polynomial
| \( x^{12} + 4 x^{11} - 6 x^{10} + 4 x^{9} + 8 x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} - 4 x^{2} + 8 x + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $32$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $4$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.2.1, 2.6.10.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: | \( x^{12} - 12 x^{11} + 74 x^{10} - 300 x^{9} + 856 x^{8} - 1760 x^{7} + 2628 x^{6} - 2840 x^{5} + 2184 x^{4} - 1152 x^{3} + 396 x^{2} - 88 x + 18 \) |
Invariants of the Galois closure
| Galois group: | $C_2^2.A_4:C_4$ (as 12T98) |
| Inertia group: | 12T55 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [2, 8/3, 8/3, 11/3, 11/3] |
| Galois mean slope: | $10/3$ |
| Galois splitting model: | $x^{12} + 24 x^{10} + 75 x^{8} - 1100 x^{6} - 1875 x^{4} + 12000 x^{2} + 3125$ |