Defining polynomial
\(x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $12$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 7 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{7}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{7}(\sqrt{3})$, 7.3.0.1, 7.4.0.1, 7.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 7.12.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3 \) |
Relative Eisenstein polynomial: | \( x - 7 \) $\ \in\Q_{7}(t)[x]$ |
Ramification polygon
The ramification polygon is trivial for unramified extensions.