Defining polynomial
|
$( x^{4} + 5 x^{2} + 4 x + 3 )^{3} + 7 x$
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{7})$ $=$$\Gal(K/\Q_{7})$: | $C_{12}$ |
| This field is Galois and abelian over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $2400 = (7^{ 4 } - 1)$ |
Intermediate fields
| $\Q_{7}(\sqrt{3})$, 7.1.3.2a1.3, 7.4.1.0a1.1, 7.2.3.4a1.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 7.4.1.0a1.1 $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{4} + 5 x^{2} + 4 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 7 t \)
$\ \in\Q_{7}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^2 + 3 z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $12$ |
| Galois group: | $C_{12}$ (as 12T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.6666666666666666$ |
| Galois splitting model: | $x^{12} - x^{11} + x^{10} - 27 x^{9} + 27 x^{8} - 183 x^{7} + 326 x^{6} + 649 x^{5} + 131 x^{4} - 573 x^{3} + 1782 x^{2} - 2133 x + 4941$ |