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.<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 <kcri...@gmail.com <javascript:>> > 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 > > > > > -- > > You received this message because you are subscribed to the Google > Groups > > "sage-devel" group. > > To unsubscribe from this group and stop receiving emails from it, send > an > > email to sage-devel+...@googlegroups.com <javascript:>. > > To post to this group, send email to sage-...@googlegroups.com > <javascript:>. > > Visit this group at https://groups.google.com/group/sage-devel. > > For more options, visit https://groups.google.com/d/optout. >
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