Defining polynomial
| \( x^{8} + 12 x^{6} + 8 x^{3} + 14 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $26$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-2*})$, 2.4.10.6 |
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: | \( x^{8} + 12 x^{6} + 8 x^{3} + 14 \) |
Invariants of the Galois closure
| Galois group: | $C_2\wr C_2^2$ (as 8T29) |
| Inertia group: | $C_2^2 \wr C_2$ |
| Unramified degree: | $2$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 3, 7/2, 4] |
| Galois mean slope: | $55/16$ |
| Galois splitting model: | $x^{8} - 16 x^{6} + 100 x^{4} - 480 x^{2} + 900$ |