An exact way to verify whether n is a congruent number using Heegner points

Abstract

We introduce the relationship between congruent numbers and elliptic curves, and compute the conductor of the elliptic curve $y^2 = x^3 - n^2 x$ associated with it. Furthermore, we prove that its $L$-series coefficient $a_m = 0$ when $m \equiv 3 \mod 4$.By using the invariants of the elliptic curve introduced above, we calculate Heegner points to quickly verify whether $n$ is a congruent number.