
Cubics and group law
(1) Let $\displaystyle C:(y^2: = x^3+ 4x)$, with the simplified group law (2.13) Show that the tangent line to $\displaystyle C$ at $\displaystyle P = (2,4)$ passes through $\displaystyle (0,0)$, and deduce that $\displaystyle P$ is a point of order 4 in the group law.
(2) Let $\displaystyle C: (y^2=x^3+ax+b)\in \mathbb{R}^2$ be nonsingular; find all points of order 2 in the group law, and understand what group they form (there are two cases to consider). Now explain geometrically how you would set about finding all points of order 4 on $\displaystyle C$.
I linked the text that we use in class above. If anyone can point me in the right direction or walk me through this, I would appreciate it. Thank you!