Defining polynomial
|
\(x^{21} + 42 x^{20} + 7 x^{19} + 42 x^{18} + 14\)
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Invariants
| Base field: | $\Q_{7}$ |
|
| Degree $d$: | $21$ |
|
| Ramification index $e$: | $21$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $38$ |
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| Discriminant root field: | $\Q_{7}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{7})$: | $C_1$ | |
| Visible Artin slopes: | $[2]$ | |
| Visible Swan slopes: | $[1]$ | |
| Means: | $\langle\frac{6}{7}\rangle$ | |
| Rams: | $(3)$ | |
| Jump set: | undefined | |
| Roots of unity: | $6 = (7 - 1)$ |
|
Intermediate fields
| 7.1.3.2a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{7}$ |
|
| Relative Eisenstein polynomial: |
\( x^{21} + 42 x^{20} + 7 x^{19} + 42 x^{18} + 14 \)
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Ramification polygon
| Residual polynomials: | $z^{14} + 3 z^7 + 3$,$3 z^6 + 2$ |
| Associated inertia: | $1$,$3$ |
| Indices of inseparability: | $[18, 0]$ |
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 |