Defining polynomial
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$( x^{6} + x^{4} + 116 x^{3} + 102 x^{2} + 3 x + 3 )^{3} + 137$
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Invariants
| Base field: | $\Q_{137}$ |
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| Degree $d$: | $18$ |
|
| Ramification index $e$: | $3$ |
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| Residue field degree $f$: | $6$ |
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| Discriminant exponent $c$: | $12$ |
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| Discriminant root field: | $\Q_{137}(\sqrt{3})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{137})$ $=$ $\Gal(K/\Q_{137})$: | $C_3\times S_3$ | |
| This field is Galois over $\Q_{137}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $6611856250608 = (137^{ 6 } - 1)$ |
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Intermediate fields
| $\Q_{137}(\sqrt{3})$, 137.3.1.0a1.1, 137.1.3.2a1.1 x3, 137.6.1.0a1.1, 137.2.3.4a1.2, 137.2.3.4a1.1 x2, 137.3.3.6a1.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 137.6.1.0a1.1 $\cong \Q_{137}(t)$ where $t$ is a root of
\( x^{6} + x^{4} + 116 x^{3} + 102 x^{2} + 3 x + 3 \)
|
|
| Relative Eisenstein polynomial: |
\( x^{3} + 137 \)
$\ \in\Q_{137}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^2 + 3 z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $18$ |
| Galois group: | $C_3\times S_3$ (as 18T3) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.6666666666666666$ |
| Galois splitting model: | not computed |