In 2025, Zywina proved that there exist infinitely many elliptic curves of rank 2 over the rationals. It is striking that such a simple fact had not been known until recently. His proof requires the polynomial Szemerédi theorem for primes, proved by Tao and Ziegler in 2008. This theorem is used to keep the number of bad primes of the elliptic curves as small as possible. The elliptic curves he obtained have exactly five bad primes. In this talk, we show that for any given set of odd primes, there exist infinitely many elliptic curves of rank 2 over the rationals whose set of bad primes contains the given set. This result is obtained by applying Zywina’s method to elliptic curves parametrized by solutions to Pell equations. Since the solutions are determined by the fundamental unit, we can arrange that the set of bad primes contains the given set.