# Cubics and group law

(1) Let $C:(y^2: = x^3+ 4x)$, with the simplified group law (2.13) Show that the tangent line to $C$ at $P = (2,4)$ passes through $(0,0)$, and deduce that $P$ is a point of order 4 in the group law.
(2) Let $C: (y^2=x^3+ax+b)\in \mathbb{R}^2$ 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 $C$.