Defining polynomial
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$( x^{4} + 8 x^{2} + 40 x + 5 )^{2} + \left(188 x^{2} + 12502\right) ( x^{4} + 8 x^{2} + 40 x + 5 ) - 15040 x^{3} + 282188 x^{2} - 1000160 x + 3390909$
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Invariants
| Base field: | $\Q_{47}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{47}$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 47 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{47}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{47}(\sqrt{5})$, $\Q_{47}(\sqrt{47})$, $\Q_{47}(\sqrt{47\cdot 5})$, 47.4.0.1, 47.4.2.1, 47.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 47.4.0.1 $\cong \Q_{47}(t)$ where $t$ is a root of
\( x^{4} + 8 x^{2} + 40 x + 5 \)
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| Relative Eisenstein polynomial: |
\( x^{2} + 47 \)
$\ \in\Q_{47}(t)[x]$
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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 |