Defining polynomial
|
\(x^{14} + x^{7} + x^{5} + x^{3} + 1\)
|
Invariants
| Base field: | $\Q_{2}$ |
|
| Degree $d$: | $14$ |
|
| Ramification index $e$: | $1$ |
|
| Residue field degree $f$: | $14$ |
|
| Discriminant exponent $c$: | $0$ |
|
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{2})$ $=$ $\Gal(K/\Q_{2})$: | $C_{14}$ | |
| This field is Galois and abelian over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | $[1]$ | |
| Roots of unity: | $32766 = (2^{ 14 } - 1) \cdot 2$ |
|
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.7.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.14.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{14} + x^{7} + x^{5} + x^{3} + 1 \)
|
|
| Relative Eisenstein polynomial: |
\( x - 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.