Generators of elliptic curves?

Question

How to find generators to group $E(\mathbb Q) $ of following elliptic curves $E:y^2=x^3-198 $, $E:y^2=x^3-122 $. Thank you in advance.

I Just used Sage software to find generators but its not giving answer. So, I need help to find generators — MKJ, Oct 25 '14 at 04:32

Jesper Petersen · Answer 1 · 2014-10-25T19:00:59.100

1

The rank of both curves are $0$. Extensive computations for all curves $y^2=x^3+k$ in the range $|k|\leq10^5$ was made by Gebel, Pethö and Zimmer in their paper On Mordell's Equation which you may find here

http://www.inf.unideb.hu/~pethoe/cikkek/67_MORDELL.pdf

The table of their results may be resurrected here

http://web.archive.org/web/20040816044914/http://emmy.math.uni-sb.de/~simath/MORDELL/MORDELL-

edited Oct 25 '14 at 19:00

answered Oct 25 '14 at 18:01

Jesper Petersen

1,128

Thank you so much for providing above link. – MKJ Oct 26 '14 at 04:42

score 0 · Answer 2 · answered Jun 15 '20 at 16:23

This is a late comment, showing that sage (at least after years) provides the correct information...

sage: E = EllipticCurve(QQ, [0, -198])
sage: E
Elliptic Curve defined by y^2 = x^3 - 198 over Rational Field
sage: E.rank(only_use_mwrank=False)
0
sage: E.torsion_order()
1

and

sage: E = EllipticCurve(QQ, [0, -122])
sage: E
Elliptic Curve defined by y^2 = x^3 - 122 over Rational Field
sage: E.rank(only_use_mwrank=False)
0
sage: E.torsion_order()
1

To make the code work, the option only_use_mwrank=False was inserted.

Generators of elliptic curves?

2 Answers2