Defining polynomial
\(x^{4} + 298 x^{3} + 22515 x^{2} + 46786 x + 3373376\) |
Invariants
Base field: | $\Q_{151}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $2$ |
Discriminant root field: | $\Q_{151}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 151 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{151}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{151}(\sqrt{3})$, $\Q_{151}(\sqrt{151})$, $\Q_{151}(\sqrt{151\cdot 3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{151}(\sqrt{3})$ $\cong \Q_{151}(t)$ where $t$ is a root of \( x^{2} + 149 x + 6 \) |
Relative Eisenstein polynomial: | \( x^{2} + 151 \) $\ \in\Q_{151}(t)[x]$ |
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: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | $x^{4} + 3473 x^{2} + 3283344$ |