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 $x+y+x \times z+y \times z= -cd$, $x \times y \times z=d$, we get the elliptic curve $[1,-c/d,-c/d,0,0]$ with a point $[0,0]$ of order $4$.

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 $4$.

Is this system of equations satisfactory for Question 1:

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

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

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 $x+y+x \times z+y \times z= -cd$, $x \times y \times z=d$, we get the elliptic curve $[1,-c/d,-c/d,0,0]$ with a point $[0,0]$ of order $4$.