Defining polynomial
| \( x^{12} - 4 x^{11} + 4 x^{10} + 4 x^{9} + 8 x^{7} + 8 x^{6} + 8 x^{4} + 8 x^{3} + 8 x^{2} + 8 x + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $32$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| 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: | \( x^{12} + 4 x^{11} - 12 x^{10} - 132 x^{9} + 496 x^{8} + 2936 x^{7} - 776 x^{6} - 36864 x^{5} + 76808 x^{4} + 117848 x^{3} - 789976 x^{2} - 1222632 x + 456642 \) |
Invariants of the Galois closure
| Galois group: | 12T193 |
| Inertia group: | 12T134 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3, 2, 3, 19/6, 19/6, 7/2] |
| Galois mean slope: | $619/192$ |
| Galois splitting model: | $x^{12} + 6 x^{10} + 9 x^{8} - 16 x^{6} - 45 x^{4} - 102 x^{2} - 49$ |