Defining polynomial
| \( x^{8} + 4 x^{7} + 4 x^{6} + 2 x^{4} + 12 x^{2} + 6 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $22$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-*})$, 2.4.8.5 |
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: | \( x^{8} + 4 x^{7} + 4 x^{6} + 2 x^{4} + 12 x^{2} + 6 \) |
Invariants of the Galois closure
| Galois group: | $C_2^2.D_4$ (as 8T19) |
| Inertia group: | $D_4\times C_2$ |
| Unramified degree: | $2$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 3, 7/2] |
| Galois mean slope: | $23/8$ |
| Galois splitting model: | $x^{8} - 4 x^{6} + 66 x^{4} + 36 x^{2} + 81$ |