# Combinations of rank and torsion attainable by $E/\mathbb{Q}$

Suppose $$r\geq 0$$ is a rank attainable by infinitely many elliptic curves over $$\mathbb{Q}$$. Let $$T$$ be one of the fifteen finite abelian groups in Mazur's theorem.

Is there an elliptic curve $$E/\mathbb{Q}$$ such that $$E(\mathbb{Q})\approx \mathbb{Z}^r\times T$$?

• I think we don't understand the rank well enough to prove anything interesting. E.g. what is the set of $r$ attainable by infinitely many curves? – WhatsUp May 3 at 10:16
• @WhatsUp I don't know what that set is but I assume $r$ is already a member of that set – keiso May 3 at 10:19
• I found a relevant paper arxiv.org/abs/2003.00077 – keiso May 3 at 10:40