Defining polynomial
\(x^{12} + 2 x^{10} + 6 x^{9} + 6 x^{8} + 8 x^{7} + 28 x^{6} + 24 x^{5} + 20 x^{4} + 24 x^{3} + 32 x^{2} + 24 x + 8\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $3$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[4/3, 4/3]$ |
Intermediate fields
2.3.0.1, 2.4.4.5 |
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: | \( x^{4} + \left(2 t^{2} + 2\right) x^{2} + 2 x + 2 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 1$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 1, 0]$ |
Invariants of the Galois closure
Galois group: | $C_3\times S_4$ (as 12T45) |
Inertia group: | Intransitive group isomorphic to $A_4$ |
Wild inertia group: | $C_2^2$ |
Unramified degree: | $6$ |
Tame degree: | $3$ |
Wild slopes: | $[4/3, 4/3]$ |
Galois mean slope: | $7/6$ |
Galois splitting model: | $x^{12} - 4 x^{9} + 54 x^{8} - 54 x^{7} + 122 x^{6} - 72 x^{5} + 93 x^{4} - 48 x^{3} + 18 x^{2} + 1$ |