Defining polynomial
|
$( x^{2} + 7 x + 2 )^{11} + \left(110 x + 55\right) ( x^{2} + 7 x + 2 )^{2} + 11$
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $22$ |
| Ramification index $e$: | $11$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $24$ |
| Discriminant root field: | $\Q_{11}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{11})$: | $C_1$ |
| This field is not Galois over $\Q_{11}.$ | |
| Visible Artin slopes: | $[\frac{6}{5}]$ |
| Visible Swan slopes: | $[\frac{1}{5}]$ |
| Means: | $\langle\frac{2}{11}\rangle$ |
| Rams: | $(\frac{1}{5})$ |
| Jump set: | undefined |
| Roots of unity: | $120 = (11^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{11}(\sqrt{2})$ |
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: |
\( x^{11} + \left(11 t + 11\right) x^{2} + 11 \)
$\ \in\Q_{11}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^2 + (6 t + 6)$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $2420$ |
| Galois group: | $C_{11}^2:C_{20}$ (as 22T18) |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | not computed |
| Galois tame degree: | not computed |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | not computed |