Defining polynomial
|
\(x^{8} + 451\)
|
Invariants
| Base field: | $\Q_{41}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $7$ |
| Discriminant root field: | $\Q_{41}(\sqrt{41\cdot 3})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{41})$ $=$ $\Gal(K/\Q_{41})$: | $C_8$ |
| This field is Galois and abelian over $\Q_{41}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $40 = (41 - 1)$ |
Intermediate fields
| $\Q_{41}(\sqrt{41\cdot 3})$, 41.1.4.3a1.3 |
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: |
\( x^{8} + 451 \)
|
Ramification polygon
| Residual polynomials: | $z^7 + 8 z^6 + 28 z^5 + 15 z^4 + 29 z^3 + 15 z^2 + 28 z + 8$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $8$ |
| Galois group: | $C_8$ (as 8T1) |
| Inertia group: | $C_8$ (as 8T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $8$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.875$ |
| Galois splitting model: | not computed |