Depending on the roots of $y^2 - x(x^2+ax+b) = 0$ being real or not, we can have 2 subgroups of points of order 2 for a given elliptic curve- Kelin-4 group or a cyclic group of order 2. How does one conclude this? Can someone elaborate on this?
Thanks, Kartik
The points on the curve with $y = 0$ correspond precisely to when the tangent line is vertical (and thus not intersecting the curve except at infinity), so that the point is of order $2$. You can see using vertical lines that if $y \ne 0$, then the inverse of a point will be the reflection across the $x$-axis.
Thus, the number of such points is determined by whether the cubic has one root or three roots, since those points will be the non-identity elements of order $2$. In the first case, the group structure must be $\Bbb Z/2\Bbb Z$. In the second case, every element is order $2$ and the subgroup is order $4$, thus it is the Klein 4 group.