Defining polynomial
| \( x^{6} - 5329 x^{2} + 5446238 \) |
Invariants
| Base field: | $\Q_{73}$ |
| Degree $d$ : | $6$ |
| Ramification exponent $e$ : | $2$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $3$ |
| Discriminant root field: | $\Q_{73}(\sqrt{73*})$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 73 })|$: | $6$ |
| This field is Galois and abelian over $\Q_{73}$. | |
Intermediate fields
| $\Q_{73}(\sqrt{73*})$, 73.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 73.3.0.1 $\cong \Q_{73}(t)$ where $t$ is a root of \( x^{3} - x + 14 \) |
| Relative Eisenstein polynomial: | $ x^{2} - 73 t \in\Q_{73}(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: | $x^{6} - x^{5} + 127 x^{4} - 127 x^{3} + 4663 x^{2} - 4663 x + 45487$ |