Defining polynomial
|
\(x^{8} + 4 x^{7} + 4 x^{2} + 8 x + 10\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[8/3, 8/3, 7/2]$ |
Intermediate fields
| 2.4.8.8 |
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: |
\( x^{8} + 4 x^{7} + 4 x^{2} + 8 x + 10 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{2} + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[15, 10, 8, 0]$ |
Invariants of the Galois closure
| Galois group: | $\GL(2,3)$ (as 8T23) |
| Inertia group: | $\SL(2,3)$ (as 8T12) |
| Wild inertia group: | $Q_8$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[8/3, 8/3, 7/2]$ |
| Galois mean slope: | $17/6$ |
| Galois splitting model: | $x^{8} + 6 x^{4} + 8 x^{2} - 3$ |