$( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + \left(20 x + 8\right) ( x^{2} + x + 1 )^{4} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$
![Copy content]()
![Toggle raw display]()
|
| Base field: | $\Q_{2}$
|
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $56$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{2})$:
|
$C_2$ |
| This field is not Galois over $\Q_{2}.$ |
| Visible Artin slopes: | $[3, \frac{7}{2}, \frac{9}{2}]$ |
| Visible Swan slopes: | $[2,\frac{5}{2},\frac{7}{2}]$ |
| Means: | $\langle1, \frac{7}{4}, \frac{21}{8}\rangle$ |
| Rams: | $(2, 3, 7)$ |
| 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: |
\( x^{8} + 8 t x^{7} + 4 t x^{6} + \left(8 t + 8\right) x^{5} + 4 t x^{4} + \left(16 t + 16\right) x^{3} + 8 t x^{2} + 8 t + 2 \)
$\ \in\Q_{2}(t)[x]$
![Copy content]()
![Toggle raw display]()
|
| Galois degree: |
$1024$
|
| Galois group: |
$C_2^4.C_2\wr C_4$ (as 16T1269)
|
| Inertia group: |
not computed
|
| Wild inertia group: |
not computed
|
| Galois unramified degree: |
$2$
|
| Galois tame degree: |
$1$
|
| Galois Artin slopes: |
$[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, 4, \frac{9}{2}]$
|
| Galois Swan slopes: |
$[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4},3,\frac{7}{2}]$
|
| Galois mean slope: |
$4.125$
|
| Galois splitting model: | not computed |
