Defining polynomial
|
\(x^{20} + 50 x^{4} + 25 x^{3} + 10\)
|
Invariants
| Base field: | $\Q_{5}$ |
|
| Degree $d$: | $20$ |
|
| Ramification index $e$: | $20$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $39$ |
|
| Discriminant root field: | $\Q_{5}(\sqrt{5\cdot 2})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{5})$: | $C_1$ | |
| Visible Artin slopes: | $[\frac{9}{4}]$ | |
| Visible Swan slopes: | $[\frac{5}{4}]$ | |
| Means: | $\langle1\rangle$ | |
| Rams: | $(5)$ | |
| Jump set: | undefined | |
| Roots of unity: | $4 = (5 - 1)$ |
|
Intermediate fields
| $\Q_{5}(\sqrt{5\cdot 2})$, 5.1.4.3a1.2 |
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: |
\( x^{20} + 50 x^{4} + 25 x^{3} + 10 \)
|
Ramification polygon
| Residual polynomials: | $z^{15} + 4 z^{10} + z^5 + 4$,$4 z^4 + 3$ |
| Associated inertia: | $1$,$4$ |
| Indices of inseparability: | $[20, 0]$ |
Invariants of the Galois closure
| Galois degree: | not computed |
| Galois group: | not computed |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | not computed |
| Galois tame degree: | not computed |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | not computed |