Defining polynomial
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\(x^{21} + 14 x^{19} + 63 x^{18} + 84 x^{17} + 756 x^{16} + 1981 x^{15} + 3813 x^{14} + 17570 x^{13} + 34825 x^{12} + 45843 x^{11} + 223244 x^{10} + 421603 x^{9} + 1580565 x^{8} + 1422905 x^{7} + 2253811 x^{6} - 3682434 x^{5} + 6238533 x^{4} + 2268805 x^{3} + 11247782 x^{2} + 6487607 x + 5926386\)
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Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $21$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $18$ |
| Discriminant root field: | $\Q_{11}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 11 }) }$: | $21$ |
| This field is Galois over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| 11.3.0.1, 11.7.6.1 x7 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 11.3.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of
\( x^{3} + 2 x + 9 \)
|
| Relative Eisenstein polynomial: |
\( x^{7} + 11 \)
$\ \in\Q_{11}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^{6} + 7z^{5} + 10z^{4} + 2z^{3} + 2z^{2} + 10z + 7$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_7:C_3$ (as 21T2) |
| Inertia group: | Intransitive group isomorphic to $C_7$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $3$ |
| Tame degree: | $7$ |
| Wild slopes: | None |
| Galois mean slope: | $6/7$ |
| Galois splitting model: | Not computed |