Defining polynomial
| \( x^{8} + 12 x^{6} + 60 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $20$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.4.6.7 |
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: | \( x^{8} - 8 x^{7} + 7126 x^{6} - 42644 x^{5} + 235210 x^{4} - 656696 x^{3} + 1489948 x^{2} - 1780136 x + 989134 \) |
Invariants of the Galois closure
| Galois group: | $C_2\wr A_4$ (as 8T38) |
| Inertia group: | $(((C_4 \times C_2): C_2):C_2):C_2$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 2, 3, 3, 7/2] |
| Galois mean slope: | $99/32$ |
| Galois splitting model: | $x^{8} - 2 x^{6} + 6 x^{4} - 6 x^{2} + 7$ |