Defining polynomial
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$( x^{8} + 3 x^{4} + 104 x^{3} + 55 x^{2} + 8 x + 3 )^{2} + 127 x$
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Invariants
| Base field: | $\Q_{127}$ |
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| Degree $d$: | $16$ |
|
| Ramification index $e$: | $2$ |
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| Residue field degree $f$: | $8$ |
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| Discriminant exponent $c$: | $8$ |
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| Discriminant root field: | $\Q_{127}(\sqrt{3})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{127})$ $=$ $\Gal(K/\Q_{127})$: | $C_{16}$ | |
| This field is Galois and abelian over $\Q_{127}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $67675234241018880 = (127^{ 8 } - 1)$ |
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Intermediate fields
| $\Q_{127}(\sqrt{3})$, 127.4.1.0a1.1, 127.8.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 127.8.1.0a1.1 $\cong \Q_{127}(t)$ where $t$ is a root of
\( x^{8} + 3 x^{4} + 104 x^{3} + 55 x^{2} + 8 x + 3 \)
|
|
| Relative Eisenstein polynomial: |
\( x^{2} + 127 t \)
$\ \in\Q_{127}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $16$ |
| Galois group: | $C_{16}$ (as 16T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $8$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.5$ |
| Galois splitting model: | not computed |