Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 125 / 3 / b

The elliptic-curve discriminant is supported at $2$ and $3$, so $5$ and $7$ are primes of good reduction. Direct point counts giveFor example, summing $1+{\chi }_p(x(x+1)(x+4))$ over $x\in {\mathbb{F}}_p$ and adding the point at infinity gives these values.

The reduction of torsion points on an elliptic curve at the two primes shows that $|E(\mathbb{Q}{)}_{\mathrm{tors}}|$ divides eight. Part (a) shows that $P_2$ has order four, and $P_1$ is an independent point of order two because it does not lie in $⟨P_2⟩$. They already generate eight points, so