Timeline for Does there exist a rational point on the elliptic curve: $y^2=x^3+6x^2+x$ ? If yes, how to find one? (relations to the 'rational distance problem') [closed]
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| S Jul 6, 2019 at 11:06 | history | suggested | Davood Khajehpour | CC BY-SA 4.0 |
better MathJax for title of the question
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| Jul 6, 2019 at 5:12 | review | Suggested edits | |||
| S Jul 6, 2019 at 11:06 | |||||
| Jun 10, 2017 at 14:56 | vote | accept | Devashish Gupta | ||
| Jun 10, 2017 at 13:54 | history | closed |
Chris Wuthrich R.P. Chris Godsil abx Henry.L |
Not suitable for this site | |
| Jun 10, 2017 at 13:44 | answer | added | Joe Silverman | timeline score: 2 | |
| Jun 10, 2017 at 13:26 | answer | added | user19475 | timeline score: 5 | |
| Jun 10, 2017 at 11:55 | comment | added | Devashish Gupta | Also, the rational solutions that we get create singularities in p. | |
| Jun 10, 2017 at 11:53 | review | Close votes | |||
| Jun 10, 2017 at 13:54 | |||||
| Jun 10, 2017 at 11:52 | comment | added | Devashish Gupta | Ok! I would surely refer to Silverman's book. | |
| Jun 10, 2017 at 11:44 | comment | added | user19475 | One can prove this result by computing $L(E,1) \neq 0$ or by doing a descent as you can find it in Silverman's book, Chapter X. | |
| Jun 10, 2017 at 11:40 | comment | added | Devashish Gupta | Wow, this implies there are no points at a rational distance on perpendicular bisector of a side of the square! | |
| Jun 10, 2017 at 11:37 | comment | added | user19475 | The only solutions are $(x,y) = (-1,-2),(0,0),(-1,2)$. There are no solutions with $x$ positive. | |
| Jun 10, 2017 at 11:35 | comment | added | Devashish Gupta | positive? (distances) | |
| Jun 10, 2017 at 11:33 | comment | added | user19475 | $x=-1, y=-2$: $(-1)^3+6(-1)^2-1 = 4 = (-2)^2$. | |
| Jun 10, 2017 at 11:20 | comment | added | Devashish Gupta | Thank you for the resources, Sir. I would be obliged, if you gave me an example of a rational point on the elliptic curve mentioned above. | |
| Jun 10, 2017 at 11:08 | comment | added | user19475 | There are formulas for point addition, see e.g. Silverman, The Arithmetic of Elliptic Curves or en.wikipedia.org/wiki/Elliptic_curve_point_multiplication. | |
| Jun 10, 2017 at 11:00 | comment | added | Devashish Gupta | How can we compute actual points using the generator? (I'm an undergraduate, 17 yrs old, but extremely interested in the problem) | |
| Jun 10, 2017 at 10:19 | review | First posts | |||
| Jun 10, 2017 at 10:23 | |||||
| Jun 10, 2017 at 10:13 | comment | added | user19475 | Magma gives that $E(\mathbf{Q}) \cong \mathbf{Z}/4$ generated by $(-1 : -2 : 1)$. | |
| Jun 10, 2017 at 10:08 | history | asked | Devashish Gupta | CC BY-SA 3.0 |