| 1 | | It seems that `isogenies_prime_degree` only finds separable isogenies, so it never finds the Frobenius. This is not documented very well. It does find the Verschiebung for ordinary elliptic curves but not for supersingular elliptic curves. |
| 2 | | |
| 3 | | Ordinary: |
| 4 | | {{{ |
| 5 | | sage: E = EllipticCurve(GF(5), [1,1]) |
| 6 | | sage: E.trace_of_frobenius() |
| 7 | | -3 |
| 8 | | sage: L = E.isogenies_prime_degree(5); L |
| 9 | | [Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5 to Elliptic Curve defined by y^2 = x^3 + x + 4 over Finite Field of size 5] |
| 10 | | }}} |
| 11 | | |
| 12 | | Supersingular: |
| 13 | | {{{ |
| 14 | | sage: E = EllipticCurve(GF(5), [0,1]) |
| 15 | | sage: E.trace_of_frobenius() |
| 16 | | 0 |
| 17 | | sage: E.isogenies_prime_degree(5) |
| 18 | | [] |
| 19 | | }}} |
| | 1 | `isogenies_prime_degree` only finds separable isogenies, so it never finds the Frobenius. This is not documented very well. It does find the Verschiebung for ordinary elliptic curves but not for supersingular elliptic curves. We clarify this in the documentation. We also add some examples and reorganize the existing examples. |
