Heegner point constructions and fundamental units in cubic fields

We prove the existence of nontrivial rational Heegner points on the elliptic curve y² = x³ + p when p ≡ 5 36 and hK is odd, and on the elliptic curve y² = x³ - p when p ≡ 23 36 and hK is odd. This follows from our expression for the fundamental unit of the non-Galois cubic extension K=Q(√[3]p) in terms of the class number hK and the norm of a special value of a modular function of level $6$ for any odd prime p ≡ 2 or 5 9, which is an analogue of a theorem of Dirichlet in 1840 expressing the fundamental unit of a real quadratic field in terms of the class number and a product of cyclotomic units.