Defining polynomial
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\(x^{14} + 2 x^{9} + x^{8} + x^{7} + 2 x^{6} + x^{5} + 2 x^{3} + x^{2} + 2\)
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Invariants
| Base field: | $\Q_{3}$ |
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| Degree $d$: | $14$ |
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| Ramification index $e$: | $1$ |
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| Residue field degree $f$: | $14$ |
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| Discriminant exponent $c$: | $0$ |
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| Discriminant root field: | $\Q_{3}(\sqrt{2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{3})$ $=$ $\Gal(K/\Q_{3})$: | $C_{14}$ | |
| This field is Galois and abelian over $\Q_{3}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $4782968 = (3^{ 14 } - 1)$ |
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Intermediate fields
| $\Q_{3}(\sqrt{2})$, $\Q_{3}(\zeta_{2186})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{3}(\zeta_{4782968})$ $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{14} + 2 x^{9} + x^{8} + x^{7} + 2 x^{6} + x^{5} + 2 x^{3} + x^{2} + 2 \)
|
|
| Relative Eisenstein polynomial: |
\( x - 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
The ramification polygon is trivial for unramified extensions.