Plane cubic curves y2=x3+Ax+B possess a group law, which has been exploited since the 1980s for factorization and cryptography. In this talk, the family of plane cubic curves given by the equation y2=x3-(t2+1)x with parameter t is considered, with particular reference to the parametrized solution x=-1, y=t. A conjecture about the relationship between this solution and the group law is supported by numerical and algebraic evidence