Defining polynomial
| \( x^{12} + 4 x^{11} + 8 x^{9} + 4 x^{4} - 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $34$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{2})$, 2.3.2.1, 2.6.11.1 |
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: | \( x^{12} - 12 x^{11} + 64 x^{10} - 200 x^{9} + 400 x^{8} - 512 x^{7} + 368 x^{6} - 32 x^{5} - 220 x^{4} + 240 x^{3} - 128 x^{2} + 32 x - 2 \) |
Invariants of the Galois closure
| Galois group: | 12T193 |
| Inertia group: | 12T134 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [2, 8/3, 8/3, 3, 7/2, 23/6, 23/6] |
| Galois mean slope: | $175/48$ |
| Galois splitting model: | $x^{12} + 22 x^{10} + 451 x^{8} + 3388 x^{6} + 9075 x^{4} + 2662 x^{2} - 14641$ |