How can I prove that the rank of $y^2=x^3+a^2x^2-a^4x$ is zero where $a$ is rational and positive?
2026-04-06 21:47:06.1775512026
On the rank of $y^2=x^3+a^2x^2-a^4x$
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For any $a\neq 0$ you can do a change of variables $x=a^2X$, $y=a^3Y$, which shows that your curve is isomorphic to $Y^2=X^3+X^2-X$, which is a curve of rank $0$.