Weierstrass form for some equation

Question

How to find a birational transformation that turns the equation $3(y^2-1)=2x^2(x^2-1)$ into Weierstrass form? Thanks!

Answer 1

First you can expand all terms and divide by $3$ to get

$$ y^2 = \frac{2}{3}x^4-\frac{2}{3}+1 $$

Then you can readily apply Theorem 6 from

http://homepage.uconn.edu/alozano/info/constrECrank_ALM.pdf

to get

$$ v^2 = u^3 - \frac{2}{3}u^2 - \frac{8}{3}u + \frac{16}{9} $$

with

$$ u = \frac{2(x-\frac{2}{3})}{y}, v = -1+\frac{u^2x}{2} $$

According to Sage the elliptic curve has rank $2$ and $2$ torsion points.