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What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.

ADDENDUM. I have just noticed that if $z^3 = x$ then there do exist some non-trivial rational point(s) provided that $t$ is a congruent number. Therefore, to avoid this and related scenarios, we impose the condition that the numerators of the reduced forms of $x \neq \pm 1$ and $z \neq \pm 1$ are relatively prime.

What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.

ADDENDUM 1. I have just noticed that if $z^3 = x$ then there do exist some non-trivial rational point(s) provided that $t$ is a congruent number. Therefore, to avoid this and related scenarios, we impose the condition that the numerators of the reduced forms of $x \neq \pm 1$ and $z \neq \pm 1$ are relatively prime.

ADDENDUM 2. It is usually interesting to generalise questions. So i would also ask for rational points on the general surface $y^2 = x^{3} -t^{2n}z^3$, $n$ being some positive integer.

What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.

ADDENDUM. I have just noticed that if $z^3 = x$ then there do exist some non-trivial rational points provided that $t$ is a congruent number. Therefore, to avoid this and related scenarios, we impose the condition that the numerators of the reduced forms of $x \neq \pm 1$ and $z \neq \pm 1$ are relatively prime.

What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.

ADDENDUM. I have just noticed that if $z^3 = x$ then there do exist some non-trivial rational point(s) provided that $t$ is a congruent number. Therefore, to avoid this and related scenarios, we impose the condition that the numerators of the reduced forms of $x \neq \pm 1$ and $z \neq \pm 1$ are relatively prime.

What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.

ADDENDUM. I have just noticed that if $z^3 = x$ then there do exist some non-trivial rational points provided that $t$ is a congruent number. Therefore, to avoid this and related scenarios, we impose the condition that the numerators of the reduced form of $x \neq \pm 1$ and $z \neq \pm 1$ are relatively prime.

What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.

ADDENDUM. I have just noticed that if $z^3 = x$ then there do exist some non-trivial rational points provided that $t$ is a congruent number. Therefore, to avoid this and related scenarios, we impose the condition that the numerators of the reduced forms of $x \neq \pm 1$ and $z \neq \pm 1$ are relatively prime.