Defining polynomial
\(x^{6} + 180 x^{5} + 10806 x^{4} + 216842 x^{3} + 32592 x^{2} + 658788 x + 13157769\) |
Invariants
Base field: | $\Q_{61}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{61}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 61 }) }$: | $6$ |
This field is Galois and abelian over $\Q_{61}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{61}(\sqrt{2})$, 61.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{61}(\sqrt{2})$ $\cong \Q_{61}(t)$ where $t$ is a root of \( x^{2} + 60 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{3} + 61 \) $\ \in\Q_{61}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + 3z + 3$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_6$ (as 6T1) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | $x^{6} - x^{5} + 62 x^{4} - 5 x^{3} + 1654 x^{2} - 720 x + 648$ |