Defining polynomial
|
\(x^{10} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 4 x + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[12/5]$ |
Intermediate fields
| 2.5.4.1 |
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: |
\( x^{10} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 4 x + 2 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$,$z^{8} + z^{6} + 1$ |
| Associated inertia: | $1$,$4$ |
| Indices of inseparability: | $[7, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\wr F_5$ (as 10T29) |
| Inertia group: | $C_2\wr C_5$ (as 10T14) |
| Wild inertia group: | $C_2^5$ |
| Unramified degree: | $4$ |
| Tame degree: | $5$ |
| Wild slopes: | $[2, 12/5, 12/5, 12/5, 12/5]$ |
| Galois mean slope: | $187/80$ |
| Galois splitting model: | $x^{10} + 2 x^{8} - 4 x^{6} + 8 x^{2} + 4$ |