$( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{5} + 12 x ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 ) + 8 x + 2$
![Copy content]()
![Toggle raw display]()
|
| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $52$ |
| 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: | $[3, \frac{7}{2}, 4]$ |
| Visible Swan slopes: | $[2,\frac{5}{2},3]$ |
| Means: | $\langle1, \frac{7}{4}, \frac{19}{8}\rangle$ |
| Rams: | $(2, 3, 5)$ |
| Jump set: | $[1, 3, 7, 15]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
![Copy content]()
![Toggle raw display]()
|
| Relative Eisenstein polynomial: |
\( x^{8} + \left(8 t + 8\right) x^{7} + 4 x^{6} + 8 x^{5} + 12 t x^{4} + 8 x^{3} + 8 t x^{2} + 8 t + 2 \)
$\ \in\Q_{2}(t)[x]$
![Copy content]()
![Toggle raw display]()
|
| Galois degree: |
$512$
|
| Galois group: |
$C_2^6.D_4$ (as 16T912)
|
| Inertia group: |
Intransitive group isomorphic to $C_2^4.D_4$
|
| Wild inertia group: |
$C_2^4.D_4$
|
| Galois unramified degree: |
$4$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 2, 3, 3, 3, \frac{7}{2}, 4]$
|
| Galois Swan slopes: |
$[1,1,2,2,2,\frac{5}{2},3]$
|
| Galois mean slope: |
$3.578125$
|
| Galois splitting model: |
$x^{16} + 8 x^{14} - 40 x^{13} + 118 x^{12} - 200 x^{11} + 956 x^{10} - 2720 x^{9} + 5185 x^{8} - 10080 x^{7} + 25796 x^{6} - 45920 x^{5} + 72938 x^{4} - 93960 x^{3} + 74828 x^{2} - 5760 x + 5751$
![Copy content]()
![Toggle raw display]()
|
