Defining polynomial
|
\(x^{4} - 4 x^{3} + 36 x^{2} + 8 x + 148\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $4$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $6$ |
| 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: | $[3]$ |
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: |
\( x^{2} + 4 t x + 8 t + 14 \)
$\ \in\Q_{2}(t)[x]$
|
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: | $[2, 3]$ |
| Galois mean slope: | $2$ |
| Galois splitting model: |
$x^{4} + 2 x^{2} - 4$
|