most recent 30 from http://mathoverflow.net 2013-05-19T17:48:01Z http://mathoverflow.net/feeds/question/60527 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60527/rank-of-subgroup-of-elliptic-curve unknown (google) 2011-04-04T07:30:56Z 2011-04-04T07:30:56Z

<p>I'm currently looking at two rational points $ p, q $ on an elliptic curve $E$ over $ \mathbb{Q} $. SAGE tells me that $E$ has rank 5 and no torsion, and that $p$ and $q$ both have infinite order. This is good news, but what I really want is for the subgroup of $E$ generated by $p, q $ to have rank 2. Is there a way for me to check that $q$ is not a multiple of $p$ (or vice versa)?</p> <p>I am extremely far from being an expert here, but somehow I have in my mind that the heights of the points will be useful. The heights of $p$ and $q$ aren't integral multiples of each other (approximately 5.5 and 5.9 respectively). Is this sufficient to show that they generate a subgroup of rank 2?</p>