Timeline for Rational points on the elliptic curve $y^2 = x^{3} - t^{2}z^3$
Current License: CC BY-SA 4.0
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| Aug 16, 2020 at 22:23 | comment | added | Gerry Myerson | 12 versions within five hours of first posting. | |
| Aug 16, 2020 at 15:35 | comment | added | Joe Silverman | (1) Please edit your question to indicate that you are looking for 4-tuples of rational numbers satisfying your equation. (2) Which "elliptic curves" are you talking about. If you plug in a value for $t$, you get a surface, not an elliptic curve. It's a rational surface, as one of the answers indicates. But probably you meant to write $y^2z=x^3-t^2z^3$, and then for $t\ne0$ you get an elliptic curve sitting in $\mathbb P^2$. These curves are related to the congruent number problem, so you'll find lots of information if you search on that term. | |
| Aug 16, 2020 at 14:07 | answer | added | Sam | timeline score: 0 | |
| Aug 12, 2020 at 5:49 | answer | added | Thomas | timeline score: 1 | |
| Aug 12, 2020 at 5:42 | history | edited | Q_p | CC BY-SA 4.0 |
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| Aug 12, 2020 at 5:29 | history | edited | Q_p | CC BY-SA 4.0 |
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| Aug 12, 2020 at 5:15 | history | edited | Q_p | CC BY-SA 4.0 |
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| Aug 12, 2020 at 3:10 | comment | added | Q_p | @R.vanDobbendeBruyn, i'm taking $t$ to be some rational number, not a formal variable, so i think what i'm looking for are $\mathbb{Q}$-rational points on the surface. | |
| Aug 12, 2020 at 3:07 | history | edited | Q_p | CC BY-SA 4.0 |
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| Aug 12, 2020 at 2:59 | comment | added | Q_p | @Kapil, i have edited the question now. | |
| Aug 12, 2020 at 2:54 | history | edited | Q_p | CC BY-SA 4.0 |
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| Aug 12, 2020 at 2:53 | comment | added | Kapil | First of all, you probably mean $y^2z$ rather than $y^2$. Secondly, from the question it appears that $t$ is allowed to be algebraic over the rationals. In that case, there are many choices of $t$ which will yield rational points. | |
| Aug 12, 2020 at 2:52 | history | edited | Q_p | CC BY-SA 4.0 |
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| Aug 12, 2020 at 2:50 | comment | added | R. van Dobben de Bruyn | But then '$t$ is a congruent number' does not make sense to me ― it is never evaluated as a number, but always functions as a formal variable... Or are you asking for $\mathbf Q$-rational points on this elliptic surface? | |
| Aug 12, 2020 at 2:46 | history | edited | Q_p | CC BY-SA 4.0 |
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| Aug 12, 2020 at 2:31 | comment | added | Will Sawin | Over $\mathbb Q(t)$? | |
| Aug 12, 2020 at 2:24 | history | edited | Q_p |
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| Aug 12, 2020 at 2:17 | history | edited | Q_p | CC BY-SA 4.0 |
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| Aug 12, 2020 at 2:11 | history | edited | Q_p | CC BY-SA 4.0 |
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| Aug 12, 2020 at 1:23 | history | edited | Q_p | CC BY-SA 4.0 |
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| Aug 12, 2020 at 1:18 | history | asked | Q_p | CC BY-SA 4.0 |