Defining polynomial
\(x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 2 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{2}.$ | |
Visible slopes: | $[3]$ |
Intermediate fields
$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{2})$, $\Q_{2}(\sqrt{2\cdot 5})$ |
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: | \( x^{2} + 14 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $C_2^2$ (as 4T2) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_2$ |
Unramified degree: | $2$ |
Tame degree: | $1$ |
Wild slopes: | $[3]$ |
Galois mean slope: | $3/2$ |
Galois splitting model: | $x^{4} - 6 x^{2} + 4$ |