How do I obtain the generator of a elliptic curve? Is E.gens() only for the generator for the free part of the group or is for torsion also?

EllipticCurve

asked 2020-01-05 10:20:33 +0100

AB
21 ●1 ●2 ●4

updated 2020-01-05 17:21:38 +0100

FrédéricC

5206 ●3 ●43 ●114

Code:

sage: E = EllipticCurve([0,-1,0,-4,-2])
sage: E.gens()    
[(3 : 2 : 1)]
sage: T = E.torsion_subgroup()
sage: T.points()
[(0 : 1 : 0), (-1 : 0 : 1)]

What is the generator of the group E(Q)?Is it the point (3,2) only? or the point (3,2)+(-1,0)?

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answered 2020-05-28 17:53:27 +0100

John Cremona
701 ●35 ●22

E.gens() gives a list of generators for E(Q) modulo torsion. You can get the generators of the torsion subgroup with E.torsion_subgroup().gens().

It would be possible to add an option to E.gens() to include torsion generators. Right now (Sage 9.1) E.gens() is a list while E.torsion_subgroup().gens() is a tuple so it is a bit awkward:

sage: E.gens()+list(E.torsion_subgroup().gens()) 
[(3 : 2 : 1), (-1 : 0 : 1)]

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Asked: 2020-01-05 10:20:33 +0100

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Last updated: May 28 '20

How do I obtain the generator of a elliptic curve? Is E.gens() only for the generator for the free part of the group or is for torsion also? edit

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