Defining polynomial
|
\(x^{20} + 5 x^{13} + 10 x^{12} + 5 x^{11} + 10\)
|
Invariants
| Base field: | $\Q_{5}$ |
|
| Degree $d$: | $20$ |
|
| Ramification index $e$: | $20$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $30$ |
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| Discriminant root field: | $\Q_{5}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{5})$: | $C_1$ | |
| Visible Artin slopes: | $[\frac{27}{16}]$ | |
| Visible Swan slopes: | $[\frac{11}{16}]$ | |
| Means: | $\langle\frac{11}{20}\rangle$ | |
| Rams: | $(\frac{11}{4})$ | |
| Jump set: | undefined | |
| Roots of unity: | $4 = (5 - 1)$ |
|
Intermediate fields
| $\Q_{5}(\sqrt{5\cdot 2})$, 5.1.4.3a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{5}$ |
|
| Relative Eisenstein polynomial: |
\( x^{20} + 5 x^{13} + 10 x^{12} + 5 x^{11} + 10 \)
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Ramification polygon
| Residual polynomials: | $z^{15} + 4 z^{10} + z^5 + 4$,$4 z + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[11, 0]$ |
Invariants of the Galois closure
| Galois degree: | not computed |
| Galois group: | not computed |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | not computed |
| Galois tame degree: | not computed |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | not computed |