Defining polynomial
| \( x^{12} + 4 x^{11} + 6 x^{10} - 4 x^{9} + 8 x^{8} - 4 x^{6} + 8 x^{5} + 8 x^{2} + 6 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $32$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.2.1, 2.6.10.6 |
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: | \( x^{12} - 12 x^{11} + 70 x^{10} - 260 x^{9} + 680 x^{8} - 1312 x^{7} + 1900 x^{6} - 2056 x^{5} + 1632 x^{4} - 928 x^{3} + 376 x^{2} - 112 x + 22 \) |
Invariants of the Galois closure
| Galois group: | $C_4^2:C_3:C_2^2$ (as 12T112) |
| Inertia group: | 12T60 |
| 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: | $x^{12} + 20 x^{10} + 153 x^{8} + 508 x^{6} + 727 x^{4} + 352 x^{2} + 121$ |