 Rational pairs of cubes summing to seven

There are infinitely many rational pairs $(u,v)$ with $u^3+v^3=7$. The corresponding Weierstrass equation of an elliptic curve is $y^2=x^3-21168$, and $(u,v)=(2,-1)$ maps to $P=(84,756)$. The primes $5$ and $11$ are good, with point counts $6$ and $12$ because cubing is bijective in their residue fields. Reduction of torsion points on an elliptic curve bounds rational torsion by order dividing $6$. The flexes of a diagonal cubic exclude three-torsion for $d=-7$, while $d/2=-7/2$ is not a cube and hence excludes two-torsion. Thus $P$ has infinite order.

All nonzero multiples of $P$ map to distinct rational affine pairs, since the only rational point at infinity of the diagonal cubic is its identity. The inverse map is $u=(y+252)/(6x)$ and $v=(252-y)/(6x)$.