Defining polynomial
|
\(x^{20} + 75 x^{4} + 5\)
|
Invariants
| Base field: | $\Q_{5}$ |
|
| Degree $d$: | $20$ |
|
| Ramification index $e$: | $20$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $39$ |
|
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{5})$ $=$ $\Gal(K/\Q_{5})$: | $F_5$ | |
| This field is Galois over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\frac{9}{4}]$ | |
| Visible Swan slopes: | $[\frac{5}{4}]$ | |
| Means: | $\langle1\rangle$ | |
| Rams: | $(5)$ | |
| Jump set: | $[1, 21]$ | |
| Roots of unity: | $20 = (5 - 1) \cdot 5$ |
|
Intermediate fields
| $\Q_{5}(\sqrt{5})$, 5.1.4.3a1.1, 5.1.5.9a1.4 x5, 5.1.10.19a2.4 x5 |
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: |
\( x^{20} + 75 x^{4} + 5 \)
|
Ramification polygon
| Residual polynomials: | $z^{15} + 4 z^{10} + z^5 + 4$,$4 z^4 + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[20, 0]$ |
Invariants of the Galois closure
| Galois degree: | $20$ |
| Galois group: | $F_5$ (as 20T5) |
| Inertia group: | $F_5$ (as 20T5) |
| Wild inertia group: | $C_5$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{9}{4}]$ |
| Galois Swan slopes: | $[\frac{5}{4}]$ |
| Galois mean slope: | $1.95$ |
| Galois splitting model: | $x^{20} + 100 x^{10} + 80$ |