Defining polynomial
\(x^{12} + 3 x + 3\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $12$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{3}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[9/8]$ |
Intermediate fields
$\Q_{3}(\sqrt{3\cdot 2})$, 3.4.3.1 |
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: | \( x^{12} + 3 x + 3 \) |
Ramification polygon
Residual polynomials: | $z + 2$,$z^{9} + z^{6} + 1$ |
Associated inertia: | $1$,$2$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $F_9:C_2$ (as 12T84) |
Inertia group: | $F_9$ (as 12T46) |
Wild inertia group: | $C_3^2$ |
Unramified degree: | $2$ |
Tame degree: | $8$ |
Wild slopes: | $[9/8, 9/8]$ |
Galois mean slope: | $79/72$ |
Galois splitting model: | $x^{12} - 6 x^{10} - 8 x^{9} + 72 x^{7} + 105 x^{6} + 24 x^{5} - 102 x^{4} - 68 x^{3} + 66 x^{2} + 36 x + 4$ |