A parametrization of θ-congruent numbers with many prime factors and with prescribed prime factors

Let θ be a real number such that 0<θ<π and cos⁡θ∈Q. For each positive integer n, we give a parametrization Sn(α) whose square-free part Nn(α) for each negative integer α is a θ-congruent number with many prime factors including any given primes (especially, at least n prime factors that are guaranteed to appear) by showing the positivity of the rank of the corresponding θ-congruent number elliptic curve over Q. Especially, we show that if a given odd prime p>2n is near 2n, then p appears as a factor of Nn(α) very often as α varies all over negative integers by proving that the probability of the set of all negative integers α such that p divides Nn(α) is 2n+1p+1.