Defining polynomial
|
\(x^{15} + 55\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $15$ |
| Ramification index $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{11})$: | $C_5$ |
| This field is not Galois over $\Q_{11}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $10 = (11 - 1)$ |
Intermediate fields
| 11.1.3.2a1.1, 11.1.5.4a1.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{11}$ |
| Relative Eisenstein polynomial: |
\( x^{15} + 55 \)
|
Ramification polygon
| Residual polynomials: | $z^{14} + 4 z^{13} + 6 z^{12} + 4 z^{11} + z^{10} + z^{3} + 4 z^{2} + 6 z + 4$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $30$ |
| Galois group: | $C_5\times S_3$ (as 15T4) |
| Inertia group: | $C_{15}$ (as 15T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $15$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[]$ |
| Galois mean slope: | $0.9333333333333333$ |
| Galois splitting model: | not computed |