$ \theta $-congruent numbers and Heegner points

Abstract.

$\theta$-congruent numbers are defined by extending congruent numbers. It has been known that a natural number n is $\theta$-congruent number if and only if the corresponding elliptic curve has positive rational rank. Using a criterion of Birch and modular parametrizations, we construct a non-trivial point on some elliptic curves by studying Heegner points on the modular curves $X_0(24)$ and $X_0(48)$.