Defining polynomial
\(x^{6} + 9 x^{2} + 12\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $11$ |
Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
Root number: | $-i$ |
$\card{ \Gal(K/\Q_{ 3 }) }$: | $6$ |
This field is Galois over $\Q_{3}.$ | |
Visible slopes: | $[5/2]$ |
Intermediate fields
$\Q_{3}(\sqrt{3\cdot 2})$, 3.3.5.2 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{3}$ |
Relative Eisenstein polynomial: | \( x^{6} + 9 x^{2} + 12 \) |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $S_3$ (as 6T2) |
Inertia group: | $S_3$ (as 6T2) |
Wild inertia group: | $C_3$ |
Unramified degree: | $1$ |
Tame degree: | $2$ |
Wild slopes: | $[5/2]$ |
Galois mean slope: | $11/6$ |
Galois splitting model: | $x^{6} - 3 x^{5} + 6 x^{4} + 35 x^{3} - 57 x^{2} - 66 x + 484$ |