Defining polynomial
|
\(x^{20} + 10 x^{19} + 15 x^{18} + 25 x^{2} + 5\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $20$ |
| Ramification index $e$: | $20$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $37$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{5})$: | $C_1$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible Artin slopes: | $[\frac{17}{8}]$ |
| Visible Swan slopes: | $[\frac{9}{8}]$ |
| Means: | $\langle\frac{9}{10}\rangle$ |
| Rams: | $(\frac{9}{2})$ |
| Jump set: | $[1, 21]$ |
| Roots of unity: | $20 = (5 - 1) \cdot 5$ |
Intermediate fields
| $\Q_{5}(\sqrt{5})$, 5.1.4.3a1.1 |
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: |
\( x^{20} + 10 x^{19} + 15 x^{18} + 25 x^{2} + 5 \)
|
Ramification polygon
| Residual polynomials: | $z^{15} + 4 z^{10} + z^5 + 4$,$4 z^2 + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[18, 0]$ |
Invariants of the Galois closure
| Galois degree: | $10000$ |
| Galois group: | $C_5^4.\OD_{16}$ (as 20T386) |
| 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 |