1 | initial version |

The group of points on an elliptic curve over a finite field is very often cyclic but may be a product of two cyclic factors, in which case the question is to give (two) generating points rather than (one) generating point. In your example,

```
sage: E1.abelian_group()
Additive abelian group isomorphic to Z/8210 embedded in Abelian group of points on Elliptic Curve defined by y^2 + x*y = x^3 + 333*x + 6166 over Finite Field of size 8209
```

shows that the group is cyclic, In any case

```
sage: E1.gens()
```

will give the list of at most two generators. For example,

```
sage: E2=EllipticCurve(GF(8209),[1,0,0,333,0])
sage: E2.gens()
[(7400 : 284 : 1), (4824 : 5797 : 1)]
```

2 | No.2 Revision |

The group of points on an elliptic curve over a finite field is very often cyclic but may be a product of two cyclic factors, in which case the question is to give (two) generating points rather than (one) generating point. In your example,

```
sage: E1.abelian_group()
Additive abelian group isomorphic to Z/8210 embedded in Abelian group of points on Elliptic Curve defined by y^2 + x*y = x^3 + 333*x + 6166 over Finite Field of size 8209
```

shows that the group is cyclic, In any case

```
sage: E1.gens()
```

will give the list of at most two generators. For example,

```
sage: E2=EllipticCurve(GF(8209),[1,0,0,333,0])
sage: E2.gens()
[(7400 : 284 : 1), (4824 : 5797 : 1)]
sage: [P.order() for P in E2.gens()]
[4062, 2]
```

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