$ \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) $ .