Defining polynomial
\(x^{12} - 1176\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{7}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 7 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{7}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{7}(\sqrt{3})$, $\Q_{7}(\sqrt{7})$, $\Q_{7}(\sqrt{7\cdot 3})$, 7.3.2.1, 7.4.2.1, 7.6.4.2, 7.6.5.3, 7.6.5.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} + 6 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{6} + 14 t + 42 \) $\ \in\Q_{7}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{5} + 6z^{4} + z^{3} + 6z^{2} + z + 6$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |