Defining polynomial
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$( x^{4} + 23 x + 6 )^{2} + \left(3772 x^{3} + 5335658 x^{2} + 3354711184 x + 791201326284\right) ( x^{4} + 23 x + 6 ) + 137543151000 x^{3} + 8953230680 x^{2} - 9808803100 x + 787157129416$
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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: |
\( x^{2} + 943 x + 41 \)
$\ \in\Q_{41}(t)[x]$
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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 |