Defining polynomial
\( x^{6} + 136 x^{3} + 7803 \) |
Invariants
Base field: | $\Q_{17}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
Root number: | $1$ |
$|\Gal(K/\Q_{ 17 })|$: | $6$ |
This field is Galois over $\Q_{17}.$ |
Intermediate fields
$\Q_{17}(\sqrt{3})$, 17.3.2.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{17}(\sqrt{3})$ $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
Relative Eisenstein polynomial: | $ x^{3} - 17 t^{3} \in\Q_{17}(t)[x]$ |
Invariants of the Galois closure
Galois group: | $S_3$ (as 6T2) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | Not computed |