Defining polynomial
|
\(x^{11} + 22 x^{6} + 11\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $11$ |
| Ramification index $e$: | $11$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $16$ |
| 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: | $[\frac{8}{5}]$ |
| Visible Swan slopes: | $[\frac{3}{5}]$ |
| Means: | $\langle\frac{6}{11}\rangle$ |
| Rams: | $(\frac{3}{5})$ |
| Jump set: | undefined |
| Roots of unity: | $10 = (11 - 1)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 11 }$. |
Canonical tower
| Unramified subfield: | $\Q_{11}$ |
| Relative Eisenstein polynomial: |
\( x^{11} + 22 x^{6} + 11 \)
|
Ramification polygon
| Residual polynomials: | $z^2 + 10$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
| Galois degree: | $55$ |
| Galois group: | $C_{11}:C_5$ (as 11T3) |
| Inertia group: | $C_{11}:C_5$ (as 11T3) |
| Wild inertia group: | $C_{11}$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $5$ |
| Galois Artin slopes: | $[\frac{8}{5}]$ |
| Galois Swan slopes: | $[\frac{3}{5}]$ |
| Galois mean slope: | $1.5272727272727273$ |
| Galois splitting model: | $x^{11} - 55 x^{9} + 1100 x^{7} - 9625 x^{5} + 34375 x^{3} - 34375 x - 12675$ |