Defining polynomial
|
\(x^{16} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 10 x^{4} + 8 x + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $38$ |
| 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: | $[\frac{4}{3}, \frac{4}{3}, 2, \frac{13}{4}]$ |
| Visible Swan slopes: | $[\frac{1}{3},\frac{1}{3},1,\frac{9}{4}]$ |
| Means: | $\langle\frac{1}{6}, \frac{1}{4}, \frac{5}{8}, \frac{23}{16}\rangle$ |
| Rams: | $(\frac{1}{3}, \frac{1}{3}, 3, 13)$ |
| Jump set: | $[1, 2, 5, 21, 37]$ |
| Roots of unity: | $4 = 2^{ 2 }$ |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.1.4.4a1.1, 2.1.8.12b1.3 |
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: |
\( x^{16} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 10 x^{4} + 8 x + 2 \)
|
Ramification polygon
| Residual polynomials: | $z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$,$1$ |
| Indices of inseparability: | $[23, 10, 4, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1536$ |
| Galois group: | $C_2^5:\GL(2,3)$ (as 16T1316) |
| 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 |