Defining polynomial
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$( x^{4} + 8 x^{2} + 54 x + 2 )^{2} + \left(268 x^{2} + 25862\right) ( x^{4} + 8 x^{2} + 54 x + 2 ) - 28944 x^{3} + 930764 x^{2} - 2793096 x + 15785669$
|
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: |
\( x^{2} + 67 \)
$\ \in\Q_{67}(t)[x]$
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Ramification polygon
Not computedInvariants 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 |