Defining polynomial
|
\(x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $20$ |
| Ramification index $e$: | $20$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $42$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_1$ |
| Visible Artin slopes: | $[\frac{38}{15}, \frac{38}{15}]$ |
| Visible Swan slopes: | $[\frac{23}{15},\frac{23}{15}]$ |
| Means: | $\langle\frac{23}{30}, \frac{23}{20}\rangle$ |
| Rams: | $(\frac{23}{3}, \frac{23}{3})$ |
| Jump set: | $[5, 15, 35]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.5.4a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 2 \)
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Ramification polygon
| Residual polynomials: | $z^{16} + z^{12} + 1$,$z + 1$ |
| Associated inertia: | $4$,$1$ |
| Indices of inseparability: | $[23, 18, 0]$ |
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 |