Defining polynomial
| \( x^{6} - 362 x^{4} + 32761 x^{2} - 1921236084 \) |
Invariants
| Base field: | $\Q_{181}$ |
| Degree $d$ : | $6$ |
| Ramification exponent $e$ : | $2$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $3$ |
| Discriminant root field: | $\Q_{181}(\sqrt{181})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 181 })|$: | $6$ |
| This field is Galois and abelian over $\Q_{181}$. | |
Intermediate fields
| $\Q_{181}(\sqrt{181})$, 181.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 181.3.0.1 $\cong \Q_{181}(t)$ where $t$ is a root of \( x^{3} - x + 18 \) |
| Relative Eisenstein polynomial: | $ x^{2} - 181 t^{2} \in\Q_{181}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $3$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | $x^{6} - x^{5} - 591 x^{4} + 682 x^{3} + 101565 x^{2} - 85050 x - 4465125$ |