Timeline for Integral points on a special curve
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 9, 2012 at 10:53 | vote | accept | user21956 | ||
| Mar 9, 2012 at 10:53 | vote | accept | user21956 | ||
| Mar 9, 2012 at 10:53 | |||||
| Mar 8, 2012 at 1:33 | answer | added | Noam D. Elkies | timeline score: 14 | |
| Mar 7, 2012 at 20:08 | comment | added | Noam D. Elkies | NB: Duje's and my posts do not contradict each other, because different versions of mwrank may (and often do) give different generators for the same group. | |
| Mar 7, 2012 at 19:57 | comment | added | James Weigandt | The high rank explains why Sage took so long. It's searching over an 8-dimensional space up to some large bound. There might be a way specific to this case to reduce the bounds up to which one needs to search. There are also some recent results on linear forms in elliptic logarithms that are not implemented in Sage. I'm not sure if these are implemented in Magma, but I do remember that Magma has a better implementation for computing integral points. At least it did at this time last year. | |
| Mar 7, 2012 at 19:35 | comment | added | duje | mw basis is generated by the points with $x$-coordinates: -695070141601562500, -1577320410156250000, -39433010253906250000, 43474893804931640625, 378951228540039062500/9, -5623634204101562500, -101469119551562500000000/994009, 415752567622704536133789062500/472932009 | |
| Mar 7, 2012 at 19:31 | comment | added | duje |
It seem that the curve comes by taking $a=25$, $b=5$ in the family of elliptic curves from this paper arxiv.org/abs/1202.5676
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| Mar 7, 2012 at 19:06 | comment | added | Noam D. Elkies | There are at least four pairs of integral points, since the mwrank basis contains points with $x = -1577320410156250000$, $43474893804931640625$, $-20492969555664062500$, and $19818207721836914062500$. There may be others. | |
| Mar 7, 2012 at 19:04 | comment | added | Noam D. Elkies | $[a_1,a_2,a_3,a_4,a_6]$ is standard notation (in this subculture) for the elliptic curve $y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$. So he's asking for integer solutions of $y^2 = x^3 - 1609983754533564186692237854003906250000 x$. Why this particular curve, I don't know. mwrank says it has rank 8. | |
| Mar 7, 2012 at 16:31 | comment | added | Vladimir Dotsenko | Care to explain your notation and motivation for what you were trying to compute? | |
| Mar 7, 2012 at 15:44 | history | asked | user21956 | CC BY-SA 3.0 |