Defining polynomial
\(x^{6} + 3 x^{5} + 3 x^{4} + 12\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $9$ |
Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
Root number: | $i$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
$\Q_{3}(\sqrt{3\cdot 2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{3}$ |
Relative Eisenstein polynomial: | \( x^{6} + 3 x^{5} + 3 x^{4} + 12 \) |
Ramification polygon
Residual polynomials: | $2z^{2} + 2$,$z^{3} + 2$ |
Associated inertia: | $2$,$1$ |
Indices of inseparability: | $[4, 0]$ |