When do Heegner points lead to a basis points?

Question

For a rank 1 elliptic curve, a rational point can be obtained from Heegner points. When is this rational point a basis point?

If sometimes additional work is required to obtain a basis point, is there an easy calculation to determine when this is the case?

Answer 1

For reference, the index of the Heegner point modulo torsion is referred to as the Heegner index. You can use the Gross-Zagier formula to numerically estimate the height of the Heegner point and thus reduce it to a finite computation by bounding the height of possible generators. This is, for example, what Sage does in its heegner_index computation.