Defining polynomial
\(x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
$\Q_{2}(\sqrt{5})$ |
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: | \( x^{2} + \left(2 t + 2\right) x + 4 t + 2 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $D_4$ (as 4T3) |
Inertia group: | Intransitive group isomorphic to $C_2^2$ |
Wild inertia group: | $C_2^2$ |
Unramified degree: | $2$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 2]$ |
Galois mean slope: | $3/2$ |
Galois splitting model: | $x^{4} - x^{2} - 1$ |