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.

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

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

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

$1$

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

$-\frac{1404928}{171}$