Defining polynomial
| \( x^{10} - 23762 x^{6} + 141158161 x^{2} - 8862473980224 \) |
Invariants
| Base field: | $\Q_{109}$ |
| Degree $d$ : | $10$ |
| Ramification exponent $e$ : | $2$ |
| Residue field degree $f$ : | $5$ |
| Discriminant exponent $c$ : | $5$ |
| Discriminant root field: | $\Q_{109}(\sqrt{109})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 109 })|$: | $10$ |
| This field is Galois and abelian over $\Q_{109}$. | |
Intermediate fields
| $\Q_{109}(\sqrt{109})$, 109.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 109.5.0.1 $\cong \Q_{109}(t)$ where $t$ is a root of \( x^{5} - x + 24 \) |
| Relative Eisenstein polynomial: | $ x^{2} - 109 t^{2} \in\Q_{109}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{10}$ (as 10T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | Not computed |