Defining polynomial
|
\(x^{22} + 262\)
|
Invariants
| Base field: | $\Q_{131}$ |
| Degree $d$: | $22$ |
| Ramification exponent $e$: | $22$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $21$ |
| Discriminant root field: | $\Q_{131}(\sqrt{131})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 131 }) }$: | $2$ |
| This field is not Galois over $\Q_{131}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{131}(\sqrt{131})$, 131.11.10.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{131}$ |
| Relative Eisenstein polynomial: |
\( x^{22} + 262 \)
|
Ramification polygon
| Residual polynomials: | $z^{21} + 22z^{20} + 100z^{19} + 99z^{18} + 110z^{17} + 3z^{16} + 74z^{15} + 113z^{14} + 130z^{13} + 13z^{12} + 30z^{11} + 128z^{10} + 30z^{9} + 13z^{8} + 130z^{7} + 113z^{6} + 74z^{5} + 3z^{4} + 110z^{3} + 99z^{2} + 100z + 22$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $D_{22}$ (as 22T3) |
| Inertia group: | $C_{22}$ (as 22T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $22$ |
| Wild slopes: | None |
| Galois mean slope: | $21/22$ |
| Galois splitting model: | Not computed |