Defining polynomial
| \( x^{12} + 2 x^{7} + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $18$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| 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: | \( x^{12} + 2 x^{7} + 2 \) |
Invariants of the Galois closure
| Galois group: | 12T254 |
| Inertia group: | 12T166 |
| Unramified degree: | $6$ |
| Tame degree: | $9$ |
| Wild slopes: | [16/9, 16/9, 16/9, 16/9, 16/9, 16/9] |
| Galois mean slope: | $127/72$ |
| Galois splitting model: | $x^{12} - 4 x^{11} - 26 x^{10} + 68 x^{9} + 494 x^{8} - 4580 x^{7} + 17380 x^{6} - 40308 x^{5} + 66036 x^{4} - 77080 x^{3} + 91480 x^{2} - 88936 x + 93500$ |