Defining polynomial
|
\(x^{8} + 284 x^{6} + 108 x^{5} + 28074 x^{4} - 13608 x^{3} + 1141144 x^{2} - 1396332 x + 15837397\)
|
Invariants
| Base field: | $\Q_{67}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{67}$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 67 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{67}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{67}(\sqrt{2})$, $\Q_{67}(\sqrt{67})$, $\Q_{67}(\sqrt{67\cdot 2})$, 67.4.0.1, 67.4.2.1, 67.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 67.4.0.1 $\cong \Q_{67}(t)$ where $t$ is a root of
\( x^{4} + 8 x^{2} + 54 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 67 \)
$\ \in\Q_{67}(t)[x]$
|
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 |