On 8 September 2017 at 12:42, John Cremona wrote:
> On 8 September 2017 at 12:28, Jeroen Demeyer wrote:
>> On 2017-09-08 11:31, John Cremona wrote:
>>>
>>> What does that look like in terms of (a,b,c)?
>>
>>
>> Totally crazy. The obvious thing gives a very complicated polynomial. The
>> problem o
On 8 September 2017 at 12:28, Jeroen Demeyer wrote:
> On 2017-09-08 11:31, John Cremona wrote:
>>
>> What does that look like in terms of (a,b,c)?
>
>
> Totally crazy. The obvious thing gives a very complicated polynomial. The
> problem of course is that everything is defined modulo the equation f
On 2017-09-08 11:31, John Cremona wrote:
What does that look like in terms of (a,b,c)?
Totally crazy. The obvious thing gives a very complicated polynomial.
The problem of course is that everything is defined modulo the equation
f(x,y,z) = 0. So you need to find a simple representative in the
On 8 September 2017 at 09:59, Jeroen Demeyer wrote:
> For completeness:
>
> We should also consider negatives. But it turns out that going from P to -P
> simply turns (a,b,c) in (b,a,c).
>
> There is also a torsion point T = (56 : 728 : 1)
>
> Adding that point gives genuinely different solutions.
We can add this as another answer there on Quora.
On Friday, September 8, 2017 at 9:47:28 AM UTC+1, John Cremona wrote:
>
> Yes, it is a nice example. Pity he does not mention Sage:
>
> sage: P2. = ProjectiveSpace(QQ,2)
> sage: f = x*(x+y)*(x+z) + y*(y+x)*(y+z) + z*(z+x)*(z+y) -
> 4*(x+y)*(y+z
For completeness:
We should also consider negatives. But it turns out that going from P to
-P simply turns (a,b,c) in (b,a,c).
There is also a torsion point T = (56 : 728 : 1)
Adding that point gives genuinely different solutions. With the torsion
point, the first solution is psi(13*P + T).
Yes, it is a nice example. Pity he does not mention Sage:
sage: P2. = ProjectiveSpace(QQ,2)
sage: f = x*(x+y)*(x+z) + y*(y+x)*(y+z) + z*(z+x)*(z+y) - 4*(x+y)*(y+z)*(z+x)
sage: phi = EllipticCurve_from_cubic(f,(-1,1,0)); phi
Scheme morphism:
From: Closed subscheme of Projective Space of dimensi
Perhaps this should be linked to on the website as well ... happy elliptic
curve reading!
https://www.quora.com/How-do-you-find-the-positive-integer-solutions-to-frac-x-y%2Bz-%2B-frac-y-z%2Bx-%2B-frac-z-x%2By-4/answer/Alon-Amit?share=1
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