Defining polynomial
| \( x^{6} - 316 \) |
Invariants
| Base field: | $\Q_{79}$ |
| Degree $d$ : | $6$ |
| Ramification exponent $e$ : | $6$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $5$ |
| Discriminant root field: | $\Q_{79}(\sqrt{79})$ |
| Root number: | $i$ |
| $|\Gal(K/\Q_{ 79 })|$: | $6$ |
| This field is Galois and abelian over $\Q_{79}$. | |
Intermediate fields
| $\Q_{79}(\sqrt{79})$, 79.3.2.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{79}$ |
| Relative Eisenstein polynomial: | \( x^{6} - 316 \) |
Invariants of the Galois closure
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | $C_6$ |
| Unramified degree: | $1$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: | $x^{6} - x^{5} - 625 x^{4} + 6810 x^{3} - 13985 x^{2} - 43097 x + 97553$ |