Yes, it is a nice example. Pity he does not mention Sage:
sage: P2.<x,y,z> = 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 dimension 2 over
Rational Field defined by:
x^3 - 3*x^2*y - 3*x*y^2 + y^3 - 3*x^2*z - 5*x*y*z - 3*y^2*z -
3*x*z^2 - 3*y*z^2 + z^3
To: Elliptic Curve defined by y^2 + 3*x*y - 91/6*y = x^3 +
43/2*x^2 - 91/4*x - 8281/216 over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(-1/6*z : y + 1/6*z : 6/91*x + 6/91*y - 1/91*z)
sage: E = phi.codomain()
sage: E
Elliptic Curve defined by y^2 + 3*x*y - 91/6*y = x^3 + 43/2*x^2 -
91/4*x - 8281/216 over Rational Field
sage: E.label()
'910c1'
sage: E.rank()
1
sage: psi = phi.inverse()
sage: P = E.gens()[0]
sage: [psi(k*P) for k in range(10)]
[(-1 : 1 : 0),
(-1/11 : 4/11 : 1),
(-8784/5165 : 9499/5165 : 1),
(-375326521/883659076 : 679733219/883659076 : 1),
(-6531563383962071/6334630576523495 : 6696085890501216/6334630576523495 : 1),
(-2798662276711559924688956/5048384306267455380784631 :
5824662475191962424632819/5048384306267455380784631 : 1),
(-399866258624438737232493646244383709/434021404091091140782000234591618320
: 287663048897224554337446918344405429/434021404091091140782000234591618320
: 1),
(-678266970930133923578916161648350398206354101381/1637627722378544613543242758851617912968156867151
:
3386928246329327259763849184510185031406211324804/1637627722378544613543242758851617912968156867151
: 1),
(-2054217703980198940765993621567260834791816664149006217306067776/2110760649231325855047088974560468667532616164397520142622104465
:
343258303254635343211175484588572430575289938927656972201563791/2110760649231325855047088974560468667532616164397520142622104465
: 1),
(36875131794129999827197811565225474825492979968971970996283137471637224634055579/4373612677928697257861252602371390152816537558161613618621437993378423467772036
:
154476802108746166441951315019919837485664325669565431700026634898253202035277999/4373612677928697257861252602371390152816537558161613618621437993378423467772036
: 1)]
sage: a,b,c = psi(10*P); a,b,c
(-893239764490691457892841485218511543529685600814390821492330243770049637905147591198476929291616899/837195266509174235125746309036231647159179965839046831731856095997939696389509603637933974969828075,
-368094076261423520198252448006319306363128641572895194683647890995604722209255509092046087984549776/837195266509174235125746309036231647159179965839046831731856095997939696389509603637933974969828075,
1)
sage: f(a,b,c)
0
sage: d=denominator(vector([a,b,c]))
sage: a, b, c = [d*t for t in a,b,c]
sage: a,b,c
(-893239764490691457892841485218511543529685600814390821492330243770049637905147591198476929291616899,
-368094076261423520198252448006319306363128641572895194683647890995604722209255509092046087984549776,
837195266509174235125746309036231647159179965839046831731856095997939696389509603637933974969828075)
sage: f(a,b,c)
0
On 8 September 2017 at 02:57, kcrisman <kcris...@gmail.com> wrote:
> 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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