Defining polynomial
|
\(x^{16} + 20 x^{10} + 2 x^{8} + 16 x^{7} + 10\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $57$ |
| Discriminant root field: | $\Q_{2}(\sqrt{2\cdot 5})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 3, 3, \frac{37}{8}]$ |
| Visible Swan slopes: | $[1,2,2,\frac{29}{8}]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{13}{8}, \frac{21}{8}\rangle$ |
| Rams: | $(1, 3, 3, 16)$ |
| Jump set: | $[1, 5, 10, 32, 48]$ |
| Roots of unity: | $4 = 2^{ 2 }$ |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{2\cdot 5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.1.4.8b1.2, 2.1.8.20d1.2 |
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: |
\( x^{16} + 20 x^{10} + 2 x^{8} + 16 x^{7} + 10 \)
|
Ramification polygon
| Residual polynomials: | $z^8 + 1$,$z^6 + 1$,$z + 1$ |
| Associated inertia: | $1$,$2$,$1$ |
| Indices of inseparability: | $[42, 26, 24, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $2048$ |
| Galois group: | $C_2^6.C_2^3.C_2^2$ (as 16T1372) |
| 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 |