Defining polynomial
|
\(x^{15} + 33\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 11 }) }$: | $5$ |
| This field is not Galois over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| 11.3.2.1, 11.5.4.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{11}$ |
| Relative Eisenstein polynomial: |
\( x^{15} + 33 \)
|
Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_5\times S_3$ (as 15T4) |
| Inertia group: | $C_{15}$ (as 15T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $15$ |
| Wild slopes: | None |
| Galois mean slope: | $14/15$ |
| Galois splitting model: | Not computed |