Defining polynomial
|
$( x^{2} + 7 x + 2 )^{4} + 44 x + 99$
|
Invariants
| Base field: | $\Q_{11}$ |
|
| Degree $d$: | $8$ |
|
| Ramification index $e$: | $4$ |
|
| Residue field degree $f$: | $2$ |
|
| Discriminant exponent $c$: | $6$ |
|
| Discriminant root field: | $\Q_{11}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{11})$ $=$ $\Gal(K/\Q_{11})$: | $Q_8$ | |
| This field is Galois 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
| $\Q_{11}(\sqrt{2})$, $\Q_{11}(\sqrt{11})$, $\Q_{11}(\sqrt{11\cdot 2})$, 11.2.2.2a1.2 |
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: |
\( x^{4} + 44 t + 99 \)
$\ \in\Q_{11}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^3 + 4 z^2 + 6 z + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $8$ |
| Galois group: | $Q_8$ (as 8T5) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.75$ |
| Galois splitting model: | $x^{8} + 44 x^{6} + 308 x^{4} + 484 x^{2} + 121$ |