Defining polynomial
\(x^{12} + x^{8} + 50 x^{7} + 53 x^{6} + 37 x^{5} + 8 x^{4} + 65 x^{3} + 103 x^{2} + 28 x + 6\)
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Invariants
Base field: | $\Q_{109}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $12$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{109}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 109 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{109}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{109}(\sqrt{2})$, 109.3.0.1, 109.4.0.1, 109.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 109.12.0.1 $\cong \Q_{109}(t)$ where $t$ is a root of
\( x^{12} + x^{8} + 50 x^{7} + 53 x^{6} + 37 x^{5} + 8 x^{4} + 65 x^{3} + 103 x^{2} + 28 x + 6 \)
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Relative Eisenstein polynomial: |
\( x - 109 \)
$\ \in\Q_{109}(t)[x]$
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Ramification polygon
The ramification polygon is trivial for unramified extensions.