Defining polynomial
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\(x^{12} + 2 x^{8} + 103 x^{7} + 83 x^{6} + 43 x^{5} + 76 x^{4} + 8 x^{3} + 177 x^{2} + x + 2\)
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Invariants
| Base field: | $\Q_{179}$ |
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| Degree $d$: | $12$ |
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| Ramification index $e$: | $1$ |
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| Residue field degree $f$: | $12$ |
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| Discriminant exponent $c$: | $0$ |
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| Discriminant root field: | $\Q_{179}(\sqrt{2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{179})$ $=$ $\Gal(K/\Q_{179})$: | $C_{12}$ | |
| This field is Galois and abelian over $\Q_{179}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $1082022699327332498100696240 = (179^{ 12 } - 1)$ |
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Intermediate fields
| $\Q_{179}(\sqrt{2})$, 179.3.1.0a1.1, 179.4.1.0a1.1, 179.6.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 179.12.1.0a1.1 $\cong \Q_{179}(t)$ where $t$ is a root of
\( x^{12} + 2 x^{8} + 103 x^{7} + 83 x^{6} + 43 x^{5} + 76 x^{4} + 8 x^{3} + 177 x^{2} + x + 2 \)
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|
| Relative Eisenstein polynomial: |
\( x - 179 \)
$\ \in\Q_{179}(t)[x]$
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Ramification polygon
The ramification polygon is trivial for unramified extensions.