Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 5 / b / Solution

The change of variables $X=5x$, $Y=25y$ identifies $E_5$ withA full two-isogeny descent, equivalently the standard full 2-descent for a cubic with three rational roots, givesThe local conditions at $2$, $5$, and infinity leave exactly these eight square-class combinations; primes outside $\{2,5\}$ have even valuations and contribute none. Hence the quotient has dimension three. Since the rational 2-torsion has dimension two, the rank is one.

The Lutz–Nagell theorem on the integral model, or reduction at two good primes, excludes odd torsion and torsion of order greater than two. Thereforeand $(-4/5,6/25)$ generates the free part. This is the Mordell-Weil group of the congruent number curve for five.