Defining polynomial
|
\(x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $56$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{20}{7}, \frac{20}{7}, \frac{20}{7}, \frac{9}{2}]$ |
| Visible Swan slopes: | $[\frac{13}{7},\frac{13}{7},\frac{13}{7},\frac{7}{2}]$ |
| Means: | $\langle\frac{13}{14}, \frac{39}{28}, \frac{13}{8}, \frac{41}{16}\rangle$ |
| Rams: | $(\frac{13}{7}, \frac{13}{7}, \frac{13}{7}, 15)$ |
| Jump set: | $[1, 3, 7, 15, 31]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.8.20a1.2 |
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: |
\( x^{16} + 4 x^{12} + 4 x^{10} + 8 x^{9} + 2 \)
|
Ramification polygon
| Residual polynomials: | $z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[41, 26, 26, 16, 0]$ |
Invariants of the Galois closure
| Galois degree: | $43008$ |
| Galois group: | $C_2^7.F_8:C_6$ (as 16T1841) |
| 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 |