\(x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 8 x^{10} + 16 x^{9} + 12 x^{8} + 16 x^{7} + 16 x^{5} + 2\)
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| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $68$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$:
|
$C_2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[3, \frac{7}{2}, \frac{9}{2}, 5]$ |
| Visible Swan slopes: | $[2,\frac{5}{2},\frac{7}{2},4]$ |
| Means: | $\langle1, \frac{7}{4}, \frac{21}{8}, \frac{53}{16}\rangle$ |
| Rams: | $(2, 3, 7, 11)$ |
| 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: | $[53, 42, 28, 16, 0]$ |
| Galois degree: |
$2048$
|
| Galois group: |
$C_2^5.C_2\wr C_2^2$ (as 16T1455)
|
| Inertia group: |
$(C_2^2\times C_4^2):\SD_{16}$ (as 16T1273)
|
| Wild inertia group: |
not computed
|
| Galois unramified degree: |
$2$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{9}{2}, \frac{9}{2}, \frac{37}{8}, 5]$
|
| Galois Swan slopes: |
$[1,1,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{7}{2},\frac{7}{2},\frac{29}{8},4]$
|
| Galois mean slope: |
$4.716796875$
|
| Galois splitting model: | $x^{16} - 8 x^{14} + 60 x^{12} - 24 x^{10} + 174 x^{8} + 168 x^{6} + 684 x^{4} + 1080 x^{2} + 729$ |
