For algebraic curves with genus 1 one can compute a number called the j invariant. An important property of this j invariant is the following: two elliptic (i.e. genus = 1) curves are birationally equivalent (i.e. can be transformed to each other with rational transformations over an algebraically closed field of constants) if and only if their j invariants are the same.

The curve must be irreducible and have genus 1, otherwise the j invariant is not defined and this procedure will fail.

$\mathrm{with}\left(\mathrm{algcurves}\right)\:$

$f≔y^5\+\frac{4}{3}-\frac{23y^2}{3}\+11y^3-\frac{17y^4}{3}-\frac{16x^2}{3}\+\frac{16x^3}{3}-\frac{4x^4}{3}\:$

$\mathrm{genus}\left(f\,x\,y\right)$

$1$

$\mathrm{j\_invariant}\left(f\,x\,y\right)$

$-\frac{1404928}{171}$