Timeline for Why is this "the first elliptic curve in nature"?
Current License: CC BY-SA 4.0
25 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2020 at 1:13 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
typo
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| Oct 30, 2020 at 1:08 | vote | accept | David Roberts♦ | ||
| Oct 30, 2020 at 1:08 | answer | added | David Roberts♦ | timeline score: 6 | |
| Oct 26, 2020 at 22:39 | vote | accept | David Roberts♦ | ||
| Oct 26, 2020 at 22:46 | |||||
| May 23, 2020 at 8:44 | comment | added | Watson | @liuyao : what you describe seems to be rather the curve lmfdb.org/EllipticCurve/Q/37/a/1, in my opinion | |
| May 20, 2020 at 23:35 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
Added extra info
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| May 20, 2020 at 15:40 | answer | added | John Cremona | timeline score: 16 | |
| May 20, 2020 at 13:46 | answer | added | Alexandre Eremenko | timeline score: 12 | |
| May 20, 2020 at 13:22 | comment | added | R.P. | If you're interested, I wrote up some things about Diophantus' treatment of (some special cases of) intersections of two quadrics in P^3 here: arxiv.org/abs/1509.06138 (section 4). Most of it is directly based on the brilliant book of Thomas Heath, only he expressed his findings purely in the language of elementary algebra. | |
| May 20, 2020 at 13:18 | comment | added | David Roberts♦ | @RP_ oh, interesting! | |
| May 20, 2020 at 13:14 | comment | added | R.P. | @DrorSpeiser Diophantus treated many more elliptic curves as well (many of his so-called 'double equations' were curves of genus 1 with a natural choice of a rational point). It would be hard to pinpoint any one of them as somehow prior to the others. | |
| May 20, 2020 at 13:07 | history | became hot network question | |||
| May 20, 2020 at 12:17 | answer | added | Nicolas Mascot | timeline score: 24 | |
| May 20, 2020 at 11:42 | comment | added | Chris Wuthrich | @FrançoisBrunault yep that is what I meant by "minimal". I believe that John's new way of enumerating curves in isogeny classes is such that the minimal one has label .a1, but I am not 100% sure. | |
| May 20, 2020 at 11:31 | comment | added | François Brunault | The elliptic curve 11a3 is the curve with smallest Faltings height, which means in basic terms that the period lattice associated to the Néron differential has largest area. I don't know if this is recorded somewhere in the literature, but it can be checked numerically at least. | |
| May 20, 2020 at 8:11 | comment | added | Chris Wuthrich | It is the minimal curve in the isogeny class with the minimal conductor. Though people.math.harvard.edu/~elkies/nature.html , which is a very nice source for elliptic curves in nature, puts it as the second and so did Cremona's tables, but it was reordered when it was taken into lmfdb. | |
| May 20, 2020 at 8:00 | comment | added | David Roberts♦ | FWIW, here is the curve considered by Diophantus: lmfdb.org/EllipticCurve/Q/8732/b/1 | |
| May 20, 2020 at 7:26 | comment | added | David Roberts♦ | This article Elliptic Curves from Mordell to Diophantus and Back by Brown and Myers in Am. Math. Monthly (doi:10.1080/00029890.2002.11919894) starts with a picture of the curve in my previous comment, labelled "The first elliptic curve". | |
| May 20, 2020 at 7:20 | comment | added | David Roberts♦ | @DrorSpeiser Well, Diophantus solved $6y - y^2 = x^3 - x$, as as example of the family $Ay - y^2 = x^3 - x$, or rather "To divide a given number [A] into two numbers such that their product is cube minus its side" (Problem IV-24; thanks to John Baez for pointing me to this) | |
| May 20, 2020 at 6:25 | comment | added | Tim Dokchitser | As far as I know, this is an informal name was coined by John Coates (and, possibly, never in writing). It is one of the three curves of smallest conductor, and has the simplest equation among those. | |
| May 20, 2020 at 6:20 | comment | added | liuyao | product of two consecutive numbers equals product of three consecutive numbers? | |
| May 20, 2020 at 6:13 | comment | added | Dror Speiser | Surely it's because it has smallest conductor. I'm guessing the congruent number curve (for 1, i.e. $y^2=x^3-x$) is the first elliptic curve in history. | |
| May 20, 2020 at 5:41 | comment | added | Robin Houston | This text seems to have been written by Nicolas Mascot, if I have interpreted LMFDB correctly; so he might be a good person to ask. lmfdb.org/knowledge/show/ec.q.11.a3.top | |
| May 20, 2020 at 4:43 | comment | added | David Roberts♦ | h/t to Anton Hilado for bringing this to my attention on Twitter! | |
| May 20, 2020 at 4:42 | history | asked | David Roberts♦ | CC BY-SA 4.0 |