Defining polynomial
|
$( x^{3} + 3 x + 3 )^{5} + \left(20 x^{2} + 10 x + 15\right) ( x^{3} + 3 x + 3 ) + 5$
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $15$ |
| Ramification index $e$: | $5$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{5})$: | $C_1$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible Artin slopes: | $[\frac{5}{4}]$ |
| Visible Swan slopes: | $[\frac{1}{4}]$ |
| Means: | $\langle\frac{1}{5}\rangle$ |
| Rams: | $(\frac{1}{4})$ |
| Jump set: | undefined |
| Roots of unity: | $124 = (5^{ 3 } - 1)$ |
Intermediate fields
| 5.3.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 5.3.1.0a1.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{3} + 3 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{5} + \left(10 t^{2} + 10 t + 20\right) x + 5 \)
$\ \in\Q_{5}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + (3 t + 4)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1500$ |
| Galois group: | $C_5^3:C_{12}$ (as 15T38) |
| Inertia group: | Intransitive group isomorphic to $C_5^3:C_4$ |
| Wild inertia group: | $C_5^3$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{5}{4}, \frac{5}{4}, \frac{5}{4}]$ |
| Galois Swan slopes: | $[\frac{1}{4},\frac{1}{4},\frac{1}{4}]$ |
| Galois mean slope: | $1.246$ |
| Galois splitting model: |
$x^{15} + 5 x^{13} - 25 x^{11} - 2 x^{10} - 100 x^{9} + 180 x^{8} + 25 x^{7} + 560 x^{6} + 102 x^{5} + 450 x^{4} - 495 x^{3} - 100 x^{2} - 300 x + 232$
|