Defining polynomial
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$( x^{2} + 192 x + 5 )^{3} + 193 x$
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Invariants
| Base field: | $\Q_{193}$ |
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| Degree $d$: | $6$ |
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| Ramification index $e$: | $3$ |
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| Residue field degree $f$: | $2$ |
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| Discriminant exponent $c$: | $4$ |
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| Discriminant root field: | $\Q_{193}(\sqrt{5})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{193})$ $=$ $\Gal(K/\Q_{193})$: | $C_6$ | |
| This field is Galois and abelian over $\Q_{193}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $37248 = (193^{ 2 } - 1)$ |
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Intermediate fields
| $\Q_{193}(\sqrt{5})$, 193.1.3.2a1.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{193}(\sqrt{5})$ $\cong \Q_{193}(t)$ where $t$ is a root of
\( x^{2} + 192 x + 5 \)
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| Relative Eisenstein polynomial: |
\( x^{3} + 193 t \)
$\ \in\Q_{193}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + 3 z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $6$ |
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.6666666666666666$ |
| Galois splitting model: |
$x^{6} - x^{5} - 837 x^{4} + 4554 x^{3} + 160855 x^{2} - 1735113 x + 4680493$
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