Defining polynomial
|
\(x^{8} + 3772 x^{7} + 5335658 x^{6} + 3354711230 x^{5} + 791201413052 x^{4} + 137665893766 x^{3} + 86143602389 x^{2} + 18207949968812 x + 5534365087156\)
|
Invariants
| Base field: | $\Q_{41}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{41}$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 41 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{41}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{41}(\sqrt{3})$, $\Q_{41}(\sqrt{41})$, $\Q_{41}(\sqrt{41\cdot 3})$, 41.4.0.1, 41.4.2.1, 41.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 41.4.0.1 $\cong \Q_{41}(t)$ where $t$ is a root of
\( x^{4} + 23 x + 6 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 943 x + 41 \)
$\ \in\Q_{41}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_4$ (as 8T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | Not computed |