Defining polynomial
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$( x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2 )^{2} + \left(-6 x^{4} - 8 x^{3} + 360 x^{2} + 10 x - 6392\right) ( x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2 ) - 1470 x^{5} + 104786 x^{4} + 22904 x^{3} - 1089868 x^{2} + 44696 x + 3555902$
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Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $6$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 11 }) }$: | $12$ |
| This field is Galois and abelian over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{11}(\sqrt{2})$, 11.3.0.1, 11.4.2.2, 11.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 11.6.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of
\( x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2 \)
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| Relative Eisenstein polynomial: |
\( x^{2} + 11 t \)
$\ \in\Q_{11}(t)[x]$
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