In Unsolved Problems in Number Theory, 3rd edition, section D1, 2004, R. K. Guy says, "Andrew Bremner has computed the rational rank of the elliptic curve x^3 + y^3 = Taxicab(n) as equal to 2, 4, 5, 4 for n = 2, 3, 4, 5, respectively." Since Taxicab(1) = 2, my question can be restated:


What is the rational rank of the elliptic curve x^3 + y^3 = Taxicab(1)?



What is the rational rank of the elliptic curve x^3 + y^3 = 2?
It's isomorphic to $y^2 = x^3 - 27$, which has rank $0$.


