Defining polynomial
|
\(x^{18} + 246\)
|
Invariants
| Base field: | $\Q_{41}$ |
| Degree $d$: | $18$ |
| Ramification index $e$: | $18$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $17$ |
| Discriminant root field: | $\Q_{41}(\sqrt{41\cdot 3})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{41})$: | $C_2$ |
| This field is not Galois over $\Q_{41}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $40 = (41 - 1)$ |
Intermediate fields
| $\Q_{41}(\sqrt{41\cdot 3})$, 41.1.3.2a1.1, 41.1.6.5a1.2, 41.1.9.8a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{41}$ |
| Relative Eisenstein polynomial: |
\( x^{18} + 246 \)
|
Ramification polygon
| Residual polynomials: | $z^{17} + 18 z^{16} + 30 z^{15} + 37 z^{14} + 26 z^{13} + 40 z^{12} + 32 z^{11} + 8 z^{10} + 11 z^9 + 35 z^8 + 11 z^7 + 8 z^6 + 32 z^5 + 40 z^4 + 26 z^3 + 37 z^2 + 30 z + 18$ |
| Associated inertia: | $6$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $108$ |
| Galois group: | $C_{18}:C_6$ (as 18T45) |
| Inertia group: | $C_{18}$ (as 18T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $18$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9444444444444444$ |
| Galois splitting model: | not computed |