If $E_1$ and $E_2$ are two elliptic curves over $\mathbb{Q}$ such that $|E_1(\mathbb{F}_p)|=|E_2(\mathbb{F}_p)|$ for all primes $p$, what does this tell us about the relationship between $E_1$ and $E_2$?
E.g. the identity holds (at least for all primes $p<500$) when $E_1$ and $E_2$ have the equations $y^2+y=x^3+x^2+2x+4$ and $y^2+y=x^3+x^2-208x-1256$. In what way are these curves similar?
They aren't isomorphic because their reduced forms ($y^2=x^3+ax+b$) are distinct. One of them has a torsion subgroup over $\mathbb{Q}$ of size 5, and the other has trivial torsion subgroup. The difference between the RHS's is $210x+1260=(2)(3)(5)(7)(x+6)$; is it relevant that this has a primorial factor? (Obviously it explains why $|E_1(\mathbb{F}_p)|=|E_2(\mathbb{F}_p)|$ for $p=2,3,5,7$, but why should this still hold for larger $p$ ??)
Many thanks for any help with this!