\(x^{16} + 8 x^{14} + 8 x^{12} + 8 x^{10} + 16 x^{9} + 2\)
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| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $72$ |
| 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: | $[3, 4, \frac{17}{4}, \frac{11}{2}]$ |
| Visible Swan slopes: | $[2,3,\frac{13}{4},\frac{9}{2}]$ |
| Means: | $\langle1, 2, \frac{21}{8}, \frac{57}{16}\rangle$ |
| Rams: | $(2, 4, 5, 15)$ |
| Jump set: | $[1, 3, 7, 15, 31]$ |
| Roots of unity: | $2$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Residual polynomials: | $z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$,$1$,$1$ |
| Indices of inseparability: | $[57, 42, 32, 16, 0]$ |
