Defining polynomial
|
\(x^{20} + 40 x^{18} + 50 x^{17} + 650 x^{16} + 1644 x^{15} + 6440 x^{14} + 18720 x^{13} + 39010 x^{12} + 137400 x^{11} + 283734 x^{10} + 677980 x^{9} + 845540 x^{8} + 1499320 x^{7} + 2045360 x^{6} + 2492700 x^{5} + 5064740 x^{4} + 1884960 x^{3} + 9207500 x^{2} - 2109440 x + 7196441\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $20$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 11 }) }$: | $20$ |
| This field is Galois and abelian over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{11}(\sqrt{2})$, 11.4.0.1, 11.5.4.4, 11.10.8.5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 11.4.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of
\( x^{4} + 8 x^{2} + 10 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{5} + 11 \)
$\ \in\Q_{11}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{4} + 5z^{3} + 10z^{2} + 10z + 5$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_{20}$ (as 20T1) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $5$ |
| Wild slopes: | None |
| Galois mean slope: | $4/5$ |
| Galois splitting model: | Not computed |