Defining polynomial
| \( x^{12} - 2 x^{10} + 4 x^{8} - 2 x^{6} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 4 x + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $24$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $4$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.3.2.1, 2.6.8.1, 2.6.10.1, 2.6.10.6 |
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: | \( x^{12} - 2 x^{10} + 4 x^{8} - 2 x^{6} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 4 x + 2 \) |
Invariants of the Galois closure
| Galois group: | $C_2\times S_4$ (as 12T23) |
| Inertia group: | $A_4 \times C_2$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [2, 8/3, 8/3] |
| Galois mean slope: | $7/3$ |
| Galois splitting model: | $x^{12} - 4 x^{11} + 12 x^{10} - 28 x^{9} + 57 x^{8} - 92 x^{7} + 114 x^{6} - 104 x^{5} + 71 x^{4} - 36 x^{3} + 14 x^{2} - 4 x + 1$ |