Defining polynomial
\(x^{12} - 8 x^{9} + 6 x^{8} - 16 x^{7} + 320 x^{6} + 448 x^{5} + 548 x^{4} - 32 x^{3} + 528 x^{2} + 248\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $24$ |
Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[2, 3]$ |
Intermediate fields
2.3.0.1, 2.6.6.6 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} + x + 1 \) |
Relative Eisenstein polynomial: | \( x^{4} + \left(6 t^{2} + 4\right) x^{2} + 4 t^{2} x + 4 t + 6 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $t^{2}z + t^{2}$,$z^{2} + t^{2}$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[5, 2, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2^3.(C_2\times A_4)$ (as 12T104) |
Inertia group: | Intransitive group isomorphic to $C_2^3:D_4$ |
Wild inertia group: | $C_2^3:D_4$ |
Unramified degree: | $3$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 2, 2, 3, 3, 3]$ |
Galois mean slope: | $91/32$ |
Galois splitting model: | $x^{12} - 39 x^{8} - 286 x^{6} - 624 x^{4} - 364 x^{2} - 13$ |