Defining polynomial
| \( x^{6} + 333 x^{3} + 34225 \) |
Invariants
| Base field: | $\Q_{37}$ |
| Degree $d$ : | $6$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $4$ |
| Discriminant root field: | $\Q_{37}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 37 })|$: | $6$ |
| This field is Galois and abelian over $\Q_{37}$. | |
Intermediate fields
| $\Q_{37}(\sqrt{*})$, 37.3.2.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{37}(\sqrt{*})$ $\cong \Q_{37}(t)$ where $t$ is a root of \( x^{2} - x + 5 \) |
| Relative Eisenstein polynomial: | $ x^{3} - 37 t^{2} \in\Q_{37}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | $x^{6} - x^{5} - 161 x^{4} + 446 x^{3} + 6467 x^{2} - 24157 x - 15107$ |