Defining polynomial
| \( x^{12} - 111 x^{9} + 4107 x^{6} - 50653 x^{3} + 14993288 \) |
Invariants
| Base field: | $\Q_{37}$ |
| Degree $d$ : | $12$ |
| Ramification exponent $e$ : | $3$ |
| Residue field degree $f$ : | $4$ |
| Discriminant exponent $c$ : | $8$ |
| Discriminant root field: | $\Q_{37}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 37 })|$: | $12$ |
| This field is Galois and abelian over $\Q_{37}$. | |
Intermediate fields
| $\Q_{37}(\sqrt{*})$, 37.3.2.1, 37.4.0.1, 37.6.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 37.4.0.1 $\cong \Q_{37}(t)$ where $t$ is a root of \( x^{4} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{3} - 37 t^{3} \in\Q_{37}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{12}$ (as 12T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Unramified degree: | $4$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | Not computed |