Defining polynomial
|
\(x^{10} + 2 x^{7} + 4 x^{3} + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $10$ |
| Ramification index $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{12}{5}]$ |
| Visible Swan slopes: | $[\frac{7}{5}]$ |
| Means: | $\langle\frac{7}{10}\rangle$ |
| Rams: | $(7)$ |
| Jump set: | $[5, 15]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.5.4a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{10} + 2 x^{7} + 4 x^{3} + 2 \)
|
Ramification polygon
| Residual polynomials: | $z^8 + z^6 + 1$,$z + 1$ |
| Associated inertia: | $4$,$1$ |
| Indices of inseparability: | $[7, 0]$ |
Invariants of the Galois closure
| Galois degree: | $320$ |
| Galois group: | $C_2^4:F_5$ (as 10T25) |
| Inertia group: | $C_2^4:C_5$ (as 10T8) |
| Wild inertia group: | $C_2^4$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $5$ |
| Galois Artin slopes: | $[\frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]$ |
| Galois Swan slopes: | $[\frac{7}{5},\frac{7}{5},\frac{7}{5},\frac{7}{5}]$ |
| Galois mean slope: | $2.3$ |
| Galois splitting model: | $x^{10} - 5 x^{8} + 10 x^{6} - 30 x^{4} + 5 x^{2} - 5$ |