Defining polynomial
| \( x^{8} + 16 x^{7} + 16 x^{4} + 80 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $24$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.4.8.6 |
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: | \( x^{8} + 130928 x^{7} - 909036 x^{6} + 2515232 x^{5} - 1790566 x^{4} - 4391672 x^{3} + 9655956 x^{2} - 7531480 x + 13518082 \) |
Invariants of the Galois closure
| Galois group: | $C_8:C_2^2$ (as 8T15) |
| Inertia group: | $Q_8:C_2$ |
| Unramified degree: | $2$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 3, 4] |
| Galois mean slope: | $25/8$ |
| Galois splitting model: | $x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 5$ |