The table presents all isomorphism classes of degree 6 extensions of F5((t)).
The data includes Defining polynomial, ramification index e, inertia degree f, valuation of discriminant c, Galois group G, Inertia Group I, Galois Slope Content.
| Defining Polynomial | e | f | c | Galois Group | #G | #I | Galois Slope Content |
|---|---|---|---|---|---|---|---|
| x^6 + x^4 + 4*x^3 + x^2 + 2 | 1 | 6 | 0 | C6 (6T1) | 6 | 1 | [ ] 1 6 |
| x^6 + 3*t^2*x^2 + 2*t^3 | 2 | 3 | 3 | C6 (6T1) | 6 | 2 | [ ] 2 3 |
| x^6 + (1 + 3*t)*x^4 + x^3 + (4 + 3*t^2)*x^2 + (3 + 2*t)*x + 4 + 4*t + 4*t^2 + t^3 | 2 | 3 | 3 | C6 (6T1) | 6 | 2 | [ ] 2 3 |
| x^6 + t*x^3 + 2*t^2 | 3 | 2 | 4 | S3 × C3 (6T5) | 18 | 3 | [ ] 3 6 |
| x^6 + 2*x^5 + 4*x^4 + (2 + 2*t)*x^3 + (3 + 2*t)*x^2 + (3 + t)*x + 3 + t^2 | 3 | 2 | 4 | S3 (6T2) | 6 | 3 | [ ] 3 2 |
| x^6 + 2*t | 6 | 1 | 5 | D6 (6T3) | 12 | 6 | [ ] 6 2 |
| x^6 + t | 6 | 1 | 5 | D6 (6T3) | 12 | 6 | [ ] 6 2 |