On the product of twists of rank two and a conjecture of Larsen

In this note we present examples of elliptic curves and infinite parametric families of pairs of integers (d,d′) such that, if we assume the parity conjecture, we can show that E ^d ,E ^d′ and E ^dd′ are all of positive even rank over ℚ. As an application, we show examples where a conjecture of M. Larsen holds.