Let $p_1=2, p_2 = 3,\ldots,$ be the prime numbers, and define $n_i = \prod_{j=1}^i p_j$. Moreover, let $E_i $ be the elliptic curve defined by $y^2 = x^3 + n_i$.

Can one compute the torsion group $E_i(\mathbb Q)_{tors}$ in terms of $i$? Or at least its cardinal?

I expect it to be trivial. I checked this for all $i\leq 1000$.

I was thinking about considering the reduction of $E_i$ modulo $p_{i+1}$ and showing that $E_i(\mathbb F_{p_{i+1}}) = \{ 0\}$. That would imply the torsion group is trivial.