Defining polynomial
\(x^{14} + 49 x^{12} + 1029 x^{10} + 12017 x^{8} + 8 x^{7} + 82859 x^{6} - 1176 x^{5} + 352947 x^{4} + 13720 x^{3} + 881203 x^{2} - 19160 x + 794999\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $7$ |
Discriminant exponent $c$: | $7$ |
Discriminant root field: | $\Q_{7}(\sqrt{7\cdot 3})$ |
Root number: | $-i$ |
$\card{ \Gal(K/\Q_{ 7 }) }$: | $14$ |
This field is Galois and abelian over $\Q_{7}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{7}(\sqrt{7\cdot 3})$, 7.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 7.7.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{7} + 6 x + 4 \) |
Relative Eisenstein polynomial: | \( x^{2} + 7 \) $\ \in\Q_{7}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |