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

Use the two-descent on an elliptic curve map associated with the three rational roots $0,-1,-4$,with the standard limiting values at the 2-torsion. Only the square classes of $-1$, $2$ and $3$ can occur, because all other numerator and denominator valuations in the three factors $x,x+1,x+4$ are even. Checking solubility over $\mathbb{R}$, ${\mathbb{Q}}_2$ and ${\mathbb{Q}}_3$ leaves exactlyThese four classes are represented respectively by $O$, $P_1$, $P_2$ and $P_1+P_2$. Thus $|E(\mathbb{Q})/2E(\mathbb{Q})|=4$. Since part (b) givesthe quotient by doubling has order $2^{r+2}$. Therefore