# How to construc a generator point of an elliptic curve?

I have taht elliptic curve defined in sage:

E1=EllipticCurve(GF(8209),[1,0,0,333,6166])

How can i construct a generator point of an elliptic curve?

Thank you very much.

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sage: E1=EllipticCurve(GF(8209),[1,0,0,333,6166])
sage: P=E1.gen(0);P
(3714 : 7019 : 1)
sage: P.order()
8210

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Thank you!!!!, i know another method but i think that hhis is better.

( 2012-06-17 19:29:35 +0200 )edit

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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