In his YouTube video [New rank records for elliptic curves having rational torsion](https://www.youtube.com/watch?v=3kxLBpj1Mzc), Noam Elkies uses systems of equations at 6:16 and 8:38 to present $\mathbb{Z}/3\mathbb{Z}$ curves of [rank 14](https://web.math.pmf.unizg.hr/%7Eduje/tors/z3old891011121314.html) and [rank 15](https://web.math.pmf.unizg.hr/%7Eduje/tors/z3.html).


I created a similar system of equations for a $\mathbb{Z}/6\mathbb{Z}$ curve of [rank 9](https://web.math.pmf.unizg.hr/%7Eduje/tors/z6.html) (edit: removed original infographic):
$$x+y+z=–73826006279$$
$$xyz=–13211438249850179974544071008000$$


I should have negated both equations, but it's too late now, as Andrej Dujella has already provided all the [integer solutions](https://twitter.com/dujella1/status/1473410623388241920).


Now I am trying to create a similar system for a curve without $3$-torsion, specifically for a $\mathbb{Z}/8\mathbb{Z}$ curve of rank 1 mentioned on p. 20 in our recently submitted paper



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> Halbeisen, Hungerbuehler, Voznyy, and Zargar, *A geometric approach to elliptic curves with torsion groups $\mathbb Z/10\mathbb Z$, $\mathbb Z/12\mathbb Z$, $\mathbb Z/14\mathbb Z$, and $\mathbb Z/16\mathbb Z$*, [arXiv: 2106.06861](https://arxiv.org/abs/2106.06861)
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$$y^2 = x^3 + 10226878x^2 + 43046721x$$



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> **Question $1$:** For the mentioned $\mathbb{Z}/8\mathbb{Z}$ curve, is it possible to keep a format similar to a $\mathbb{Z}/3\mathbb{Z}$ case (two equations, three variables, +, -, $\times$)? I would really like to avoid squaring/cubing the same variable.
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> **Question $2$:** Is it possible to keep a similar format for any other curve having a $2$-torsion, but not $3$-torsion, i.e. for torsion subgroups $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/8\mathbb{Z}$, $\mathbb{Z}/10\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}$?
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Systems of equations for elliptic curves without $3$-torsion
Is this system of equations satisfactory for Question 1:


$x+y+x \times z+y \times z=6400$,


$x \times y \times z=6561$?


It seems that all curves with torsion groups containing $\mathbb{Z}/4\mathbb{Z}$ can be obtained in this way. By taking $x+y+x \times z+y \times z= d$, $x \times y \times z=-cd$, we get the elliptic curve $[1,-c/d,-c/d,0,0]$ with a point $[0,0]$ of order $4$.


