\(x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 8 x^{12} + 8 x^{10} + 8 x^{4} + 10\)
![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: |
$1536$
|
| Galois group: |
$C_2^6:D_{12}$ (as 16T1313)
|
| Inertia group: |
$C_2^6:C_{12}$ (as 16T1041)
|
| Wild inertia group: |
$C_2^6:C_4$
|
| Galois unramified degree: |
$2$
|
| Galois tame degree: |
$3$
|
| Galois Artin slopes: |
$[\frac{4}{3}, \frac{4}{3}, 3, \frac{19}{6}, \frac{19}{6}, 4, \frac{49}{12}, \frac{49}{12}]$
|
| Galois Swan slopes: |
$[\frac{1}{3},\frac{1}{3},2,\frac{13}{6},\frac{13}{6},3,\frac{37}{12},\frac{37}{12}]$
|
| Galois mean slope: |
$3.9244791666666665$
|
| Galois splitting model: |
$x^{16} - 8 x^{15} - 16 x^{14} + 208 x^{13} + 76 x^{12} - 2072 x^{11} + 360 x^{10} + 6736 x^{9} - 2310 x^{8} + 6840 x^{7} - 17568 x^{6} + 1728 x^{5} + 17220 x^{4} - 13304 x^{3} + 7400 x^{2} - 1232 x + 179$
![Copy content]()
![Toggle raw display]()
|
