Defining polynomial
|
\(x^{13} + 6 x^{2} + 4\)
|
Invariants
| Base field: | $\Q_{7}$ |
|
| Degree $d$: | $13$ |
|
| Ramification index $e$: | $1$ |
|
| Residue field degree $f$: | $13$ |
|
| Discriminant exponent $c$: | $0$ |
|
| Discriminant root field: | $\Q_{7}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{7})$ $=$ $\Gal(K/\Q_{7})$: | $C_{13}$ | |
| This field is Galois and abelian over $\Q_{7}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $96889010406 = (7^{ 13 } - 1)$ |
|
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$. |
Canonical tower
| Unramified subfield: | $\Q_{7}(\zeta_{96889010406})$ $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{13} + 6 x^{2} + 4 \)
|
|
| Relative Eisenstein polynomial: |
\( x - 7 \)
$\ \in\Q_{7}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.