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I have a bivariate modular polynomial of shape $$f(x,y)=x^2y-g(x)\equiv 0\bmod q$$ where $q=(2p-1)(2p+1)$ is a product of two primes $2p-1$ and $2p+1$, $g(x)\in\mathbb Z[x]$ is of degree four and $f(...

I need an algorithm to efficiently find at least one lattice point on a hyperbola of equation $axy+bx+cy+d=0$. Lattice point means integer coordinates and equation with integer means diophantine ...