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algcurves

 j_invariant
 The j invariant of an elliptic curve

 Calling Sequence j_invariant(f, x, y)

Parameters

 f - polynomial in x and y representing a curve of genus 1 x, y - variables

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{algcurves}\right):$
 > $f≔{y}^{5}+\frac{4}{3}-\frac{23{y}^{2}}{3}+11{y}^{3}-\frac{17{y}^{4}}{3}-\frac{16{x}^{2}}{3}+\frac{16{x}^{3}}{3}-\frac{4{x}^{4}}{3}:$

Check that the genus is 1, because only then is the j invariant defined.

 > $\mathrm{genus}\left(f,x,y\right)$
 ${1}$ (1)
 > $\mathrm{j_invariant}\left(f,x,y\right)$
 ${-}\frac{{1404928}}{{171}}$ (2)