Defining polynomial
|
$( x^{2} + 42 x + 3 )^{3} + \left(86 x + 1806\right) ( x^{2} + 42 x + 3 ) + 150672 x + 3165961$
|
Invariants
| Base field: | $\Q_{43}$ |
| Degree $d$: | $6$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{43}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 43 }) }$: | $6$ |
| This field is Galois and abelian over $\Q_{43}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{43}(\sqrt{2})$, 43.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{43}(\sqrt{2})$ $\cong \Q_{43}(t)$ where $t$ is a root of
\( x^{2} + 42 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 43 \)
$\ \in\Q_{43}(t)[x]$
|
Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: |
$x^{6} - x^{5} + 218 x^{4} + 296 x^{3} + 11584 x^{2} + 7168 x + 32768$
|