Defining polynomial
| \( x^{8} + 20 x^{2} + 12 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $14$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-*})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.4.4.5 |
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: | \( x^{8} + 30 x^{7} + 522 x^{6} + 5568 x^{5} + 39960 x^{4} + 177336 x^{3} + 458226 x^{2} + 330048 x + 207846 \) |
Invariants of the Galois closure
| Galois group: | $C_2^3:S_4.C_2$ (as 8T44) |
| Inertia group: | $C_2\wr A_4$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [4/3, 4/3, 2, 7/3, 7/3, 5/2] |
| Galois mean slope: | $223/96$ |
| Galois splitting model: | $x^{8} + 20 x^{2} + 12$ |