I'm looking for some kind of 3-dimensional Elliptic curve. As far as I know a normal Elliptic curve like $$y^2 = x^3 + ax + b$$ over $F_p$ consists of one or two cyclic groups $Z_m (\times Z_n)$. Given one point and one/two generators you can reach each point with it. Now I'm looking for a product of 3 cyclic groups, given a point and 3 generators let you reach each point in this product. (btw for cryptography do you mainly use elliptic curves with one ore two cyclic groups?)
So lets say it's $Z_m \times Z_n \times Z_r$ with generators $g_m, g_n, g_r$ and a point $s$ which lies in there. With $s g_m^i g_n^j g_r^k$ all points can be reached. E.g. $s g_m^m$ would be $s$ again and $s g_m^i g_n^j = s g_n^j g_m^i$ (same for $r$,$m$,$n$). (Best for use case would be $m=n=r=prim$)
I read from Wikipedia about Projective twisted Edwards coordinates with $$(a X^2 + Y^2)Z^2 = Z^4 + d X^2 Y^2$$
It has 3 coordinates but not sure if it's what I'm searching for. Can't find information about the inner structure. I did some tests with the written algorithm but it was neither cyclic for all points nor closed for all points nor a product of 3 cyclic groups. I did some test with small numbers ($p,a,d$) and found many different cycle sizes. Normal for that kind of Elliptic curves? Did I something wrong? Is it what I'm looking for?
Do you know any other Elliptic curve with an inner structure of 3 cyclic groups (some more also Ok, can ignore those)? Or something else which produces 3 cyclic groups, with the condition, given two points, starting at one you don't know how to reach the other point. Only way doing this try around until finding him. So you don't know which generator using next. And if you found him you also know the way backward.