Defining polynomial
|
\(x^{11} + 77 x + 11\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $11$ |
| Ramification exponent $e$: | $11$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $11$ |
| Discriminant root field: | $\Q_{11}(\sqrt{11})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 11 }) }$: | $1$ |
| This field is not Galois over $\Q_{11}.$ | |
| Visible slopes: | $[11/10]$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 11 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{11}$ |
| Relative Eisenstein polynomial: |
\( x^{11} + 77 x + 11 \)
|
Ramification polygon
| Residual polynomials: | $z + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $F_{11}$ (as 11T4) |
| Inertia group: | $F_{11}$ (as 11T4) |
| Wild inertia group: | $C_{11}$ |
| Unramified degree: | $1$ |
| Tame degree: | $10$ |
| Wild slopes: | $[11/10]$ |
| Galois mean slope: | $119/110$ |
| Galois splitting model: | Not computed |