How to compute the order of (2, 3) on the elliptic curve y^2 = x^3 + 1?

Question

Since $y^2 = x^3 + 1 $ has a group structure, how could you compute this by Mordell–Weil theorem? Thank you!

Answer 1

The tangent to the curve at $P_1 = (2,3)$ is $y = 2 x - 1$. This intersects the curve at $(0,-1)$, so $2 P_1 = (0,1)$. The line through $2 P_1$ and $P_1$ is $y = x + 1$, which intersects the curve at $(-1,0)$, so $3 P_1 = (-1,0)$. The line tangent to the curve at $3 P_1$ is $x=-1$, which does not intersect the curve anywhere else. Thus $6 P_1$ is the "point at infinity" $O$, and the order of $P_1$ is $6$.