Defining polynomial
| \( x^{8} + 4 x^{6} + 12 x^{4} + 16 x^{3} + 16 x^{2} + 16 x + 14 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $29$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-2})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{2})$, 2.4.10.3 |
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: | \( x^{8} + 4 x^{6} + 12 x^{4} + 16 x^{3} + 16 x^{2} + 16 x + 14 \) |
Invariants of the Galois closure
| Galois group: | $C_4^2:C_4$ (as 8T30) |
| Inertia group: | $(((C_4 \times C_2): C_2):C_2):C_2$ |
| Unramified degree: | $1$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 3, 7/2, 4, 17/4, 19/4] |
| Galois mean slope: | $137/32$ |
| Galois splitting model: | $x^{8} + 8 x^{4} - 16 x^{2} + 56$ |