Defining polynomial
|
\(x^{20} + 148\)
|
Invariants
| Base field: | $\Q_{37}$ |
| Degree $d$: | $20$ |
| Ramification index $e$: | $20$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $19$ |
| Discriminant root field: | $\Q_{37}(\sqrt{37})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{37})$: | $C_4$ |
| This field is not Galois over $\Q_{37}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $36 = (37 - 1)$ |
Intermediate fields
| $\Q_{37}(\sqrt{37})$, 37.1.4.3a1.3, 37.1.5.4a1.1, 37.1.10.9a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{37}$ |
| Relative Eisenstein polynomial: |
\( x^{20} + 148 \)
|
Ramification polygon
| Residual polynomials: | $z^{19} + 20 z^{18} + 5 z^{17} + 30 z^{16} + 35 z^{15} + z^{14} + 21 z^{13} + 5 z^{12} + 22 z^{11} + 17 z^{10} + 15 z^9 + 17 z^8 + 22 z^7 + 5 z^6 + 21 z^5 + z^4 + 35 z^3 + 30 z^2 + 5 z + 20$ |
| Associated inertia: | $4$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $80$ |
| Galois group: | $C_4\times F_5$ (as 20T20) |
| Inertia group: | $C_{20}$ (as 20T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $20$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.95$ |
| Galois splitting model: | not computed |