Defining polynomial
|
\(x^{16} + 2 x^{15} + 4 x^{12} + 4 x^{9} + 4 x^{5} + 4 x^{3} + 2 x^{2} + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $30$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{11}{4}]$ |
| Visible Swan slopes: | $[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{7}{4}]$ |
| Means: | $\langle\frac{1}{14}, \frac{3}{28}, \frac{1}{8}, \frac{15}{16}\rangle$ |
| Rams: | $(\frac{1}{7}, \frac{1}{7}, \frac{1}{7}, 13)$ |
| Jump set: | $[1, 2, 4, 9, 25]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.8.8a1.1 |
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: |
\( x^{16} + 2 x^{15} + 4 x^{12} + 4 x^{9} + 4 x^{5} + 4 x^{3} + 2 x^{2} + 2 \)
|
Ramification polygon
| Residual polynomials: | $z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[15, 2, 2, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $21504$ |
| Galois group: | $C_2^7:F_8:C_3$ (as 16T1800) |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | not computed |
| Galois tame degree: | not computed |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | not computed |