Defining polynomial
|
$( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{3} + 6$
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $36$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_2\times C_4$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 2, 3]$ |
| Visible Swan slopes: | $[1,1,2]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{4}, \frac{11}{8}\rangle$ |
| Rams: | $(1, 1, 5)$ |
| Jump set: | $[1, 3, 11, 19]$ |
| Roots of unity: | $12 = (2^{ 2 } - 1) \cdot 2^{ 2 }$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-5})$, 2.2.2.4a1.1, 2.2.2.4a2.1 x2, 2.1.4.6a1.2 x2, 2.2.4.12a1.3, 2.2.4.16b2.2, 2.2.4.16b2.6 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{8} + 4 x^{7} + 2 x^{6} + 4 t x^{3} + 6 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^6 + 1$,$z + t$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[11, 6, 6, 0]$ |
Invariants of the Galois closure
| Galois degree: | $32$ |
| Galois group: | $C_2^2.D_4$ (as 16T54) |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 3, 3]$ |
| Galois Swan slopes: | $[1,1,2,2]$ |
| Galois mean slope: | $2.625$ |
| Galois splitting model: | not computed |