Defining polynomial
|
\(x^{16} + 8 x^{15} + 8 x^{10} + 8 x^{4} + 16 x^{3} + 10\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $62$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_1$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[3, 4, \frac{17}{4}, \frac{17}{4}]$ |
| Visible Swan slopes: | $[2,3,\frac{13}{4},\frac{13}{4}]$ |
| Means: | $\langle1, 2, \frac{21}{8}, \frac{47}{16}\rangle$ |
| Rams: | $(2, 4, 5, 5)$ |
| Jump set: | $[1, 3, 7, 15, 31]$ |
| Roots of unity: | $2$ |
Intermediate fields
| $\Q_{2}(\sqrt{-2\cdot 5})$, 2.1.4.11a1.6 |
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: |
\( x^{16} + 8 x^{15} + 8 x^{10} + 8 x^{4} + 16 x^{3} + 10 \)
|
Ramification polygon
| Residual polynomials: | $z^8 + 1$,$z^4 + 1$,$z^3 + z + 1$ |
| Associated inertia: | $1$,$1$,$3$ |
| Indices of inseparability: | $[47, 42, 32, 16, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1536$ |
| Galois group: | $C_2^6:(C_2\times A_4)$ (as 16T1299) |
| 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 |