Defining polynomial
| \( x^{10} + 143 x^{5} + 5929 \) |
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$ : | $10$ |
| Ramification exponent $e$ : | $5$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $8$ |
| Discriminant root field: | $\Q_{11}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 11 })|$: | $10$ |
| This field is Galois and abelian over $\Q_{11}$. | |
Intermediate fields
| $\Q_{11}(\sqrt{*})$, 11.5.4.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{11}(\sqrt{*})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} - x + 7 \) |
| Relative Eisenstein polynomial: | $ x^{5} - 11 t^{2} \in\Q_{11}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{10}$ (as 10T1) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Unramified degree: | $2$ |
| Tame degree: | $5$ |
| Wild slopes: | None |
| Galois mean slope: | $4/5$ |
| Galois splitting model: | $x^{10} - x^{9} + 2 x^{8} + 326 x^{7} - 536 x^{6} + 4816 x^{5} + 43381 x^{4} - 339673 x^{3} + 908642 x^{2} - 1781528 x + 1928783$ |