Defining polynomial
|
\(x^{18} + 66603 x^{6} - 486268503\)
|
Invariants
| Base field: | $\Q_{149}$ |
| Degree $d$: | $18$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{149}(\sqrt{149\cdot 2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 149 }) }$: | $6$ |
| This field is not Galois over $\Q_{149}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{149}(\sqrt{149\cdot 2})$, 149.3.0.1, 149.3.2.1, 149.6.3.2, 149.6.5.2, 149.9.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 149.3.0.1 $\cong \Q_{149}(t)$ where $t$ is a root of
\( x^{3} + 3 x + 147 \)
|
| Relative Eisenstein polynomial: |
\( x^{6} + 149 t \)
$\ \in\Q_{149}(t)[x]$
|
Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_6\times S_3$ (as 18T6) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: | Not computed |