\(x^{16} + 8 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{9} + 4 x^{8} + 8 x^{4} + 16 x^{3} + 2\)
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| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $66$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$:
|
$C_2^2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[3, 4, \frac{17}{4}, \frac{19}{4}]$ |
| Visible Swan slopes: | $[2,3,\frac{13}{4},\frac{15}{4}]$ |
| Means: | $\langle1, 2, \frac{21}{8}, \frac{51}{16}\rangle$ |
| Rams: | $(2, 4, 5, 9)$ |
| 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: | $[51, 42, 32, 16, 0]$ |
| Galois degree: |
$128$
|
| Galois group: |
$C_2^5:C_4$ (as 16T259)
|
| Inertia group: |
$C_2\wr C_4$ (as 16T157)
|
| Wild inertia group: |
$C_2\wr C_4$
|
| Galois unramified degree: |
$2$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$
|
| Galois Swan slopes: |
$[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]$
|
| Galois mean slope: |
$4.28125$
|
| Galois splitting model: | $x^{16} - 16 x^{14} - 32 x^{13} + 36 x^{12} + 384 x^{11} + 848 x^{10} - 800 x^{9} - 6034 x^{8} - 6656 x^{7} + 6960 x^{6} + 23072 x^{5} + 21684 x^{4} + 7232 x^{3} - 2032 x^{2} - 2272 x - 527$ |
