Defining polynomial
|
\(x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $20$ |
| Ramification exponent $e$: | $10$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $18$ |
| Discriminant root field: | $\Q_{11}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 11 }) }$: | $20$ |
| This field is Galois and abelian over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{11}(\sqrt{2})$, $\Q_{11}(\sqrt{11})$, $\Q_{11}(\sqrt{11\cdot 2})$, 11.4.2.1, 11.5.4.4, 11.10.8.5, 11.10.9.1, 11.10.9.7 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of
\( x^{2} + 7 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{10} + 11 \)
$\ \in\Q_{11}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{9} + 10z^{8} + z^{7} + 10z^{6} + z^{5} + 10z^{4} + z^{3} + 10z^{2} + z + 10$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_{10}$ (as 20T3) |
| Inertia group: | Intransitive group isomorphic to $C_{10}$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $10$ |
| Wild slopes: | None |
| Galois mean slope: | $9/10$ |
| Galois splitting model: | Not computed |