Defining polynomial
\(x^{11} + 66 x + 11\)  |
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: | $-i$ |
| $|\Aut(K/\Q_{ 11 })|$: | $1$ |
| This field is not Galois over $\Q_{11}.$ | |
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: | \( x^{11} + 66 x + 11 \) |
Invariants of the Galois closure
| Galois group: | $F_{11}$ (as 11T4) |
| Inertia group: | $F_{11}$ |
| Unramified degree: | $1$ |
| Tame degree: | $10$ |
| Wild slopes: | [11/10] |
| Galois mean slope: | $119/110$ |
| Galois splitting model: | Not computed |