Defining polynomial
|
\(x^{20} + 5 x^{16} + 120\)
|
Invariants
| Base field: | $\Q_{5}$ |
|
| Degree $d$: | $20$ |
|
| Ramification index $e$: | $20$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $35$ |
|
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{5})$ $=$ $\Gal(K/\Q_{5})$: | $C_{20}$ | |
| This field is Galois and abelian over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[2]$ | |
| Visible Swan slopes: | $[1]$ | |
| Means: | $\langle\frac{4}{5}\rangle$ | |
| Rams: | $(4)$ | |
| Jump set: | undefined | |
| Roots of unity: | $4 = (5 - 1)$ |
|
Intermediate fields
| $\Q_{5}(\sqrt{5})$, 5.1.4.3a1.4, 5.1.5.8a4.1, 5.1.10.17a3.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{5}$ |
|
| Relative Eisenstein polynomial: |
\( x^{20} + 5 x^{16} + 120 \)
|
Ramification polygon
| Residual polynomials: | $z^{15} + 4 z^{10} + z^5 + 4$,$4 z^4 + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[16, 0]$ |