Defining polynomial
|
\(x^{16} + 2 x^{12} + 4 x^{11} + 2 x^{8} + 8 x^{4} + 4 x^{2} + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $42$ |
| 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: | $[2, 2, \frac{5}{2}, \frac{13}{4}]$ |
| Visible Swan slopes: | $[1,1,\frac{3}{2},\frac{9}{4}]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{4}, \frac{9}{8}, \frac{27}{16}\rangle$ |
| Rams: | $(1, 1, 3, 9)$ |
| Jump set: | $[1, 2, 7, 15, 31]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.4.6a2.1, 2.1.8.16c2.1 |
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: |
\( x^{16} + 2 x^{12} + 4 x^{11} + 2 x^{8} + 8 x^{4} + 4 x^{2} + 2 \)
|
Ramification polygon
| Residual polynomials: | $z^{12} + z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $3$,$1$,$1$ |
| Indices of inseparability: | $[27, 18, 12, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $6144$ |
| Galois group: | $C_4^4:(C_2\times A_4)$ (as 16T1657) |
| 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 |