Defining polynomial
\(x^{6} + 393 x^{5} + 51492 x^{4} + 2250723 x^{3} + 208317 x^{2} + 7054242 x + 307845740\) |
Invariants
Base field: | $\Q_{137}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{137}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 137 }) }$: | $6$ |
This field is Galois over $\Q_{137}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{137}(\sqrt{3})$, 137.3.2.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{137}(\sqrt{3})$ $\cong \Q_{137}(t)$ where $t$ is a root of \( x^{2} + 131 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{3} + 137 \) $\ \in\Q_{137}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + 3z + 3$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $S_3$ (as 6T2) |
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: | Not computed |