= Solution
The change of variables $X=5x$, $Y=25y$ identifies $E_5$ with
$$
Y^2=X^3-25X=X(X-5)(X+5).
$$
A full <two-isogeny descent>, equivalently the standard full 2-descent for a cubic with three rational roots, gives
$$
E_5(\mathbb Q)/2E_5(\mathbb Q)
=\langle(0,0),(1,0),(-4/5,6/25)\rangle_{\mathbb F_2}.
$$
The 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. Therefore
$$
E_5(\mathbb Q)_{\mathrm{tors}}=\{O,(0,0),(1,0),(-1,0)\}\cong(\mathbb Z/2\mathbb Z)^2,
$$
and $(-4/5,6/25)$ generates the free part. This is the <Mordell-Weil group of the congruent number curve for five>.
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