Defining polynomial
| \( x^{12} + 4 x^{10} + 4 x^{9} + 4 x^{8} + 6 x^{6} + 4 x^{3} - 6 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $26$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $4$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{2*})$, $\Q_{2}(\sqrt{-2*})$, 2.3.2.1, 2.4.8.1, 2.6.8.1, 2.6.11.5, 2.6.11.13 |
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: | \( x^{12} + 4 x^{10} + 4 x^{9} + 4 x^{8} + 6 x^{6} + 4 x^{3} + 10 \) |
Invariants of the Galois closure
| Galois group: | $C_2^2\times S_3$ (as 12T10) |
| Inertia group: | $C_6\times C_2$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [2, 3] |
| Galois mean slope: | $13/6$ |
| Galois splitting model: | $x^{12} - 4 x^{11} + 16 x^{9} - 6 x^{8} - 80 x^{7} + 198 x^{6} - 140 x^{5} - 38 x^{4} - 148 x^{3} + 528 x^{2} - 448 x + 122$ |