Defining polynomial
\(x^{12} + 10 x^{11} + 47 x^{10} + 144 x^{9} + 330 x^{8} + 578 x^{7} + 785 x^{6} + 830 x^{5} + 530 x^{4} - 64 x^{3} - 189 x^{2} - 30 x + 25\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $16$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 2 }) }$: | $12$ |
This field is Galois over $\Q_{2}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-5})$, 2.3.2.1 x3, 2.4.4.1, 2.6.4.1, 2.6.8.1 x3, 2.6.8.3 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} + x + 1 \) |
Relative Eisenstein polynomial: | \( x^{6} + 2 x^{5} + 2 x^{3} + 2 x^{2} + 6 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{4} + z^{2} + 1$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
Galois group: | $D_6$ (as 12T3) |
Inertia group: | Intransitive group isomorphic to $C_6$ |
Wild inertia group: | $C_2$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | $[2]$ |
Galois mean slope: | $4/3$ |
Galois splitting model: | $x^{12} - 3 x^{10} + 2 x^{8} + x^{6} + 2 x^{4} - 3 x^{2} + 1$ |