Defining polynomial
| \( x^{12} + 4 x^{11} - 4 x^{10} - 4 x^{9} + 4 x^{8} + 8 x^{7} + 4 x^{6} + 4 x^{4} + 8 x + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $12$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $32$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-*})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-2})$, 2.3.2.1, 2.6.11.9 |
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: | \( x^{12} - 12 x^{11} + 108 x^{10} - 956 x^{9} + 8988 x^{8} - 64968 x^{7} + 328356 x^{6} - 1137680 x^{5} + 2411988 x^{4} - 1874592 x^{3} - 3364768 x^{2} + 8621736 x - 5994270 \) |
Invariants of the Galois closure
| Galois group: | 12T193 |
| Inertia group: | 12T134 |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3, 2, 3, 19/6, 19/6, 7/2] |
| Galois mean slope: | $619/192$ |
| Galois splitting model: | $x^{12} - 44 x^{10} - 84 x^{9} + 518 x^{8} + 2032 x^{7} + 1188 x^{6} - 5080 x^{5} - 9100 x^{4} - 1456 x^{3} + 8792 x^{2} + 8344 x + 2402$ |