Defining polynomial
| \( x^{12} + 4 x^{11} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 4 x + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $24$ |
| 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: | \( x^{12} + 4 x^{11} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 4 x + 2 \) |
Invariants of the Galois closure
| Galois group: | 12T254 |
| Inertia group: | 12T166 |
| Unramified degree: | $6$ |
| Tame degree: | $9$ |
| Wild slopes: | [22/9, 22/9, 22/9, 22/9, 22/9, 22/9] |
| Galois mean slope: | $697/288$ |
| Galois splitting model: | $x^{12} - 8 x^{11} + 16 x^{10} + 64 x^{9} - 950 x^{8} + 6072 x^{7} - 16800 x^{6} + 1296 x^{5} + 83788 x^{4} - 82896 x^{3} - 293280 x^{2} + 700128 x - 442152$ |