Timeline for Integral points in elliptic curves
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 17, 2017 at 2:39 | comment | added | Joe Silverman | Your equation $y^2=x^3−39x^22+320x+1024$ is not a minimal Weierstrass equation, it is not minimal at 2. If I'm reading the output from PARI correctly, a change of variables to get a minimal equation is $x\to 4x$ and $y\to 8y+4x$, although it requires another change of variables to put it in the form listed in the LMFDB. In any case, I guess the answer to the question you're really asking is that the integer points on the minimal W.E. may not give all integral pts on your curve, since you also need pts on the minimal W.E. with $x=a/4$ and $y=b/8$ with $a$ and $b$ integers. | |
| Oct 17, 2017 at 0:20 | comment | added | Zlatko | How exactly is this Weierstrass model computed (which transformation is used)? | |
| Oct 6, 2017 at 13:18 | comment | added | Joe Silverman | ------ That's a good question. I don't know what algorithm is being used to find integral points. However, for curves of rank 1, I know that the methods used are pretty good, and can often be used to prove that the list is complete (via linear forms in elliptic logs, for example). You might try looking at the LMFDB documentation to see if they explain how reliable their lists of integral points are. | |
| Oct 6, 2017 at 13:16 | comment | added | Joe Silverman | From: "Zoran Stanic" <[email protected]> Date: Fri, 06 Oct 2017 14:55:52 +0200 Dear Professor Silverman, Sorry for interrupting you directly, but unfortunately I am new on MathOverflow, so I cannot add a comment. About your answer to my question regarding Integral points in elliptic curves: Are you sure that ALL integral points are listed (up to reversing the sign) for that particular or any other curve? (In other words, would I be noticed if there is a possible existence of other int. points?) | |
| Oct 5, 2017 at 17:16 | history | edited | Joe Silverman | CC BY-SA 3.0 |
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| Oct 5, 2017 at 15:21 | history | answered | Joe Silverman | CC BY-SA 3.0 |