I have been working on a problem which involves a cubic equation in two variables of the form:

3(ny^2-(n-2)y)=nx^3 3x^2-(n-3)x

n can be any nonzero positive integer. I am finding to find integral solutions to curves of this form. Simple Python does pretty well at getting a lot of solutions but I'd like to get a deeper understanding of the problem by using Sage and its many useful built-in operations. My curve seems like it's an elliptic curve but sage requires all elliptic curves to be written in the form:

y^2 axy cy=x^3 bx^2 dx f

It would be nice if my curve could be expressed in this form somehow but the problem is the factor of 3 that is on the left side of the equation. If I divide it out, I get a factor of 1/3 on the cubic term on the other side. Is there any way to do this? I will accept written solutions as well as coding solutions in python/sage. My math proficiency is pretty good but I was not a math major so I'll need an explanation for technical terms that don't appear in physics