Defining polynomial
| \( x^{12} + 2 x^{11} + 2 x^{9} + 2 x^{6} + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $20$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $1$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.3.2.1 |
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: | \( x^{12} + 2 x^{11} + 2 x^{9} + 2 x^{6} + 2 \) |
Invariants of the Galois closure
| Galois group: | 12T206 |
| Inertia group: | 12T90 |
| Unramified degree: | $6$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3, 4/3, 4/3, 2, 2] |
| Galois mean slope: | $175/96$ |
| Galois splitting model: | $x^{12} - 2 x^{11} - 116 x^{10} + 1090 x^{9} - 4240 x^{8} + 14780 x^{7} - 59578 x^{6} + 115320 x^{5} - 130784 x^{4} + 257960 x^{3} + 81096 x^{2} + 164724 x + 85590$ |