Defining polynomial
|
\(x^{12} - 444 x^{9} + 54760 x^{6} + 27960456 x^{3} + 7496644\)
|
Invariants
| Base field: | $\Q_{37}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{37}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 37 }) }$: | $12$ |
| This field is Galois and abelian over $\Q_{37}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{37}(\sqrt{2})$, 37.3.2.3, 37.4.0.1, 37.6.4.3 |
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
\( x^{4} + 6 x^{2} + 24 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 37 t^{2} \)
$\ \in\Q_{37}(t)[x]$
|
Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_{12}$ (as 12T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | Not computed |