Defining polynomial
|
$( x^{3} + x + 145 )^{4} + 151 x$
|
Invariants
| Base field: | $\Q_{151}$ |
|
| Degree $d$: | $12$ |
|
| Ramification index $e$: | $4$ |
|
| Residue field degree $f$: | $3$ |
|
| Discriminant exponent $c$: | $9$ |
|
| Discriminant root field: | $\Q_{151}(\sqrt{151\cdot 3})$ | |
| Root number: | $i$ | |
| $\Aut(K/\Q_{151})$: | $C_6$ | |
| This field is not Galois over $\Q_{151}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $3442950 = (151^{ 3 } - 1)$ |
|
Intermediate fields
| $\Q_{151}(\sqrt{151})$, 151.3.1.0a1.1, 151.1.4.3a1.2, 151.3.2.3a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 151.3.1.0a1.1 $\cong \Q_{151}(t)$ where $t$ is a root of
\( x^{3} + x + 145 \)
|
|
| Relative Eisenstein polynomial: |
\( x^{4} + 151 t \)
$\ \in\Q_{151}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^3 + 4 z^2 + 6 z + 4$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $24$ |
| Galois group: | $C_3\times D_4$ (as 12T14) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.75$ |
| Galois splitting model: | not computed |