\(x^{16} + 8 x^{15} + 8 x^{13} + 8 x^{10} + 10\)
![Copy content]()
![Toggle raw display]()
|
| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $60$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$:
|
$C_1$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[3, 4, \frac{49}{12}, \frac{49}{12}]$ |
| Visible Swan slopes: | $[2,3,\frac{37}{12},\frac{37}{12}]$ |
| Means: | $\langle1, 2, \frac{61}{24}, \frac{45}{16}\rangle$ |
| Rams: | $(2, 4, \frac{13}{3}, \frac{13}{3})$ |
| 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.
| Galois degree: |
$3072$
|
| Galois group: |
$C_2^6:(C_2\times S_4)$ (as 16T1519)
|
| Inertia group: |
$C_2^6:(C_2\times A_4)$ (as 16T1299)
|
| Wild inertia group: |
not computed
|
| Galois unramified degree: |
$2$
|
| Galois tame degree: |
$3$
|
| Galois Artin slopes: |
$[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4, \frac{49}{12}, \frac{49}{12}]$
|
| Galois Swan slopes: |
$[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3,\frac{37}{12},\frac{37}{12}]$
|
| Galois mean slope: |
$3.9309895833333335$
|
| Galois splitting model: |
$x^{16} - 8 x^{15} + 24 x^{14} + 16 x^{13} - 152 x^{12} - 4320 x^{10} + 3456 x^{9} + 29880 x^{8} - 101280 x^{7} + 137760 x^{6} - 22464 x^{5} - 285696 x^{4} + 540288 x^{3} - 435456 x^{2} + 149184 x - 17352$
![Copy content]()
![Toggle raw display]()
|
