Defining polynomial
|
\(x^{20} + 2928\)
|
Invariants
| Base field: | $\Q_{61}$ |
|
| Degree $d$: | $20$ |
|
| Ramification index $e$: | $20$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $19$ |
|
| Discriminant root field: | $\Q_{61}(\sqrt{61})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{61})$ $=$ $\Gal(K/\Q_{61})$: | $C_{20}$ | |
| This field is Galois and abelian over $\Q_{61}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $60 = (61 - 1)$ |
|
Intermediate fields
| $\Q_{61}(\sqrt{61})$, $\Q_{61}(\sqrt[4]{61})$, $\Q_{61}(\sqrt[5]{61})$, $\Q_{61}(\sqrt[10]{61})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{61}$ |
|
| Relative Eisenstein polynomial: |
\( x^{20} + 2928 \)
|
Ramification polygon
| Residual polynomials: | $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $20$ |
| Galois group: | $C_{20}$ (as 20T1) |
| Inertia group: | $C_{20}$ (as 20T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $20$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.95$ |
| Galois splitting model: | not computed |