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In his YouTube video New rank records for elliptic curves having rational torsion, Noam Elkies uses systems of equations at 6:16 and 8:38 to present $\mathbb{Z}/3\mathbb{Z}$ curves of rank 14 and rank 15.

I created a similar system of equations for a $\mathbb{Z}/6\mathbb{Z}$ curve of rank 9 (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.

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

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

$$y^2 = x^3 + 10226878x^2 + 43046721x$$

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.

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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    $\begingroup$ To make this self containd, could you explain what "fruit equations" are? $\endgroup$
    – LSpice
    Commented Dec 22, 2021 at 4:12
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    $\begingroup$ They are equations where variables are substituted by fruit emojis. Some recent examples: link 1, link 2, and link 3. $\endgroup$
    – Maksym Voznyy
    Commented Dec 22, 2021 at 4:21

1 Answer 1

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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 $4$.

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    $\begingroup$ Following the suggestion in the answer, so far I was able to show that for the curves with torsion groups $\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/8\mathbb{Z}$ and presented as $[0,a_2,0,a_4,0]$, it is possible to write $(x+y)(z+1)=2\sqrt{a_2+2\sqrt{a_4}}$ and $xyz=\sqrt{a_4}$. $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}$ don't follow suit yet. $\endgroup$
    – Maksym Voznyy
    Commented Dec 23, 2021 at 4:06
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    $\begingroup$ For all curves with torsion groups containing $\mathbb{Z}/4\mathbb{Z}$, by comparing the $j$-invariants for $[0,a_2,0,a_4,0]$ and $[1,-c/d,-c/d,0,0]$, the possible values of $r=-c/d$ are obtained by solving the degree $6$ equation $\frac{-(1-16r+16r^2)^3}{r^4(-1+16r)}=\frac{256(a_2^2-3a_4)^3}{a_4^2(a_2^2-4a_4)}$. Then, for each torsion group, it is possible to write at least one system of equations $(x+y)(z+1)=d$, $xyz=-cd$. $\endgroup$
    – Maksym Voznyy
    Commented Dec 23, 2021 at 17:43

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