Defining polynomial
\(x^{6} - 16002 x^{3} + 48387\) |
Invariants
Base field: | $\Q_{127}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{127}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 127 }) }$: | $6$ |
This field is Galois and abelian over $\Q_{127}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{127}(\sqrt{3})$, 127.3.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{127}(\sqrt{3})$ $\cong \Q_{127}(t)$ where $t$ is a root of \( x^{2} + 126 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{3} + 127 t \) $\ \in\Q_{127}(t)[x]$ |
Ramification polygon
Data not computedInvariants 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: | $x^{6} - x^{5} + 2 x^{4} + 4378 x^{3} + 130035 x^{2} - 721815 x + 2868703$ |