Defining polynomial
|
$( x^{2} + 7 x + 2 )^{10} + 110 x + 33$
|
Invariants
| Base field: | $\Q_{11}$ |
|
| Degree $d$: | $20$ |
|
| Ramification index $e$: | $10$ |
|
| Residue field degree $f$: | $2$ |
|
| Discriminant exponent $c$: | $18$ |
|
| Discriminant root field: | $\Q_{11}(\sqrt{2})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{11})$ $=$ $\Gal(K/\Q_{11})$: | $C_{20}$ | |
| This field is Galois and abelian over $\Q_{11}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $120 = (11^{ 2 } - 1)$ |
|
Intermediate fields
| $\Q_{11}(\sqrt{2})$, 11.2.2.2a1.1, 11.1.5.4a1.1, 11.2.5.8a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of
\( x^{2} + 7 x + 2 \)
|
|
| Relative Eisenstein polynomial: |
\( x^{10} + 110 t + 33 \)
$\ \in\Q_{11}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^9 + 10 z^8 + z^7 + 10 z^6 + z^5 + 10 z^4 + z^3 + 10 z^2 + z + 10$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $20$ |
| Galois group: | $C_{20}$ (as 20T1) |
| Inertia group: | Intransitive group isomorphic to $C_{10}$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $10$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9$ |
| Galois splitting model: | not computed |