Rational Points of Projective Curves: X/Y+Y/Z+Z/X = n (n \in [0..100])

[2005.01.05]X/Y+Y/Z+Z/X=n(n \in [0..100])の有理点

• x^2+xy^2+y=6xyのグラフ(jpeg)
• x^2+xy^2+y=6xyのグラフ(jpeg):原点付近
• y^2+6xy+y=x^3のグラフ(jpeg)
• y^2+6xy+y=x^3のグラフ(jpeg):原点付近
• x/y+y/z+z/x=nの有理点を計算するプログラム(pari/gp)

gp>  en=ec(n)
time = 12 ms.
%1 = [n, 0, 1, 0, 0, n^2, n, 1, 0, n^4 - 24*n, -n^6 + 36*n^3 - 216, n^3 - 27, (n^12 - 72*n^9 + 1728*n^6 - 13824*n^3)/(n^3 - 27), 0, 0, 0, 0, 0, 0]
gp>  en.disc
time = 0 ms.
%2 = n^3 - 27
gp>  factor(en.disc)
time = 54 ms.
%3 = 
[n - 3 1]

[n^2 + 3*n + 9 1]

gp>  T=[0,0]
time = 0 ms.
%4 = [0, 0]
gp>  ellpow(en,T,2)
time = 0 ms.
%5 = [0, -1]
gp>  ellpow(en,T,3)
time = 0 ms.
%6 = [0]

gp>  for(n=0,100,if(n!=3,print("E_",n,"_{tors}=",elltors(ec(n),1))))
E_0_{tors}=[3, [3], [[0, 0]]]
E_1_{tors}=[3, [3], [[0, 0]]]
E_2_{tors}=[3, [3], [[0, 0]]]
E_4_{tors}=[3, [3], [[0, 0]]]
E_5_{tors}=[6, [6], [[-2, 8]]]
E_6_{tors}=[3, [3], [[0, 0]]]
E_7_{tors}=[3, [3], [[0, 0]]]
E_8_{tors}=[3, [3], [[0, 0]]]
E_9_{tors}=[3, [3], [[0, 0]]]
E_10_{tors}=[3, [3], [[0, 0]]]
E_11_{tors}=[3, [3], [[0, 0]]]
E_12_{tors}=[3, [3], [[0, 0]]]
E_13_{tors}=[3, [3], [[0, 0]]]
E_14_{tors}=[3, [3], [[0, 0]]]
E_15_{tors}=[3, [3], [[0, 0]]]
E_16_{tors}=[3, [3], [[0, 0]]]
E_17_{tors}=[3, [3], [[0, 0]]]
E_18_{tors}=[3, [3], [[0, 0]]]
E_19_{tors}=[3, [3], [[0, 0]]]
E_20_{tors}=[3, [3], [[0, 0]]]
E_21_{tors}=[3, [3], [[0, 0]]]
E_22_{tors}=[3, [3], [[0, 0]]]
E_23_{tors}=[3, [3], [[0, 0]]]
E_24_{tors}=[3, [3], [[0, 0]]]
E_25_{tors}=[3, [3], [[0, 0]]]
E_26_{tors}=[3, [3], [[0, 0]]]
E_27_{tors}=[3, [3], [[0, 0]]]
E_28_{tors}=[3, [3], [[0, 0]]]
E_29_{tors}=[3, [3], [[0, 0]]]
E_30_{tors}=[3, [3], [[0, 0]]]
E_31_{tors}=[3, [3], [[0, 0]]]
E_32_{tors}=[3, [3], [[0, 0]]]
E_33_{tors}=[3, [3], [[0, 0]]]
E_34_{tors}=[3, [3], [[0, 0]]]
E_35_{tors}=[3, [3], [[0, 0]]]
E_36_{tors}=[3, [3], [[0, 0]]]
E_37_{tors}=[3, [3], [[0, 0]]]
E_38_{tors}=[3, [3], [[0, 0]]]
E_39_{tors}=[3, [3], [[0, 0]]]
E_40_{tors}=[3, [3], [[0, 0]]]
E_41_{tors}=[3, [3], [[0, 0]]]
E_42_{tors}=[3, [3], [[0, 0]]]
E_43_{tors}=[3, [3], [[0, 0]]]
E_44_{tors}=[3, [3], [[0, 0]]]
E_45_{tors}=[3, [3], [[0, 0]]]
E_46_{tors}=[3, [3], [[0, 0]]]
E_47_{tors}=[3, [3], [[0, 0]]]
E_48_{tors}=[3, [3], [[0, 0]]]
E_49_{tors}=[3, [3], [[0, 0]]]
E_50_{tors}=[3, [3], [[0, 0]]]
E_51_{tors}=[3, [3], [[0, 0]]]
E_52_{tors}=[3, [3], [[0, 0]]]
E_53_{tors}=[3, [3], [[0, 0]]]
E_54_{tors}=[3, [3], [[0, 0]]]
E_55_{tors}=[3, [3], [[0, 0]]]
E_56_{tors}=[3, [3], [[0, 0]]]
E_57_{tors}=[3, [3], [[0, 0]]]
E_58_{tors}=[3, [3], [[0, 0]]]
E_59_{tors}=[3, [3], [[0, 0]]]
E_60_{tors}=[3, [3], [[0, 0]]]
E_61_{tors}=[3, [3], [[0, 0]]]
E_62_{tors}=[3, [3], [[0, 0]]]
E_63_{tors}=[3, [3], [[0, 0]]]
E_64_{tors}=[3, [3], [[0, 0]]]
E_65_{tors}=[3, [3], [[0, 0]]]
E_66_{tors}=[3, [3], [[0, 0]]]
E_67_{tors}=[3, [3], [[0, 0]]]
E_68_{tors}=[3, [3], [[0, 0]]]
E_69_{tors}=[3, [3], [[0, 0]]]
E_70_{tors}=[3, [3], [[0, 0]]]
E_71_{tors}=[3, [3], [[0, 0]]]
E_72_{tors}=[3, [3], [[0, 0]]]
E_73_{tors}=[3, [3], [[0, 0]]]
E_74_{tors}=[3, [3], [[0, 0]]]
E_75_{tors}=[3, [3], [[0, 0]]]
E_76_{tors}=[3, [3], [[0, 0]]]
E_77_{tors}=[3, [3], [[0, 0]]]
E_78_{tors}=[3, [3], [[0, 0]]]
E_79_{tors}=[3, [3], [[0, 0]]]
E_80_{tors}=[3, [3], [[0, 0]]]
E_81_{tors}=[3, [3], [[0, 0]]]
E_82_{tors}=[3, [3], [[0, 0]]]
E_83_{tors}=[3, [3], [[0, 0]]]
E_84_{tors}=[3, [3], [[0, 0]]]
E_85_{tors}=[3, [3], [[0, 0]]]
E_86_{tors}=[3, [3], [[0, 0]]]
E_87_{tors}=[3, [3], [[0, 0]]]
E_88_{tors}=[3, [3], [[0, 0]]]
E_89_{tors}=[3, [3], [[0, 0]]]
E_90_{tors}=[3, [3], [[0, 0]]]
E_91_{tors}=[3, [3], [[0, 0]]]
E_92_{tors}=[3, [3], [[0, 0]]]
E_93_{tors}=[3, [3], [[0, 0]]]
E_94_{tors}=[3, [3], [[0, 0]]]
E_95_{tors}=[3, [3], [[0, 0]]]
E_96_{tors}=[3, [3], [[0, 0]]]
E_97_{tors}=[3, [3], [[0, 0]]]
E_98_{tors}=[3, [3], [[0, 0]]]
E_99_{tors}=[3, [3], [[0, 0]]]
E_100_{tors}=[3, [3], [[0, 0]]]
time = 1,150 ms.
gp>  e=ec(5);for(i=2,6,print(i,"[-2,8]=",ellpow(e,[-2,8],i)))
2[-2,8]=[0, -1]
3[-2,8]=[-1/4, 1/8]
4[-2,8]=[0, 0]
5[-2,8]=[-2, 1]
6[-2,8]=[0]
time = 6 ms.

bash-2.05a$ mwrank3
Program mwrank: uses 2-descent (via 2-isogeny if possible) to
determine the rank of an elliptic curve E over Q, and list a
set of points which generate E(Q) modulo 2E(Q).
and finally search for further points on the curve.
For more details see the file mwrank.doc.
For details of algorithms see the author's book.

Please acknowledge use of this program in published work, 
and send problems to John.Cremona@nottingham.ac.uk.

Version compiled on Feb 11 2003 at 17:40:15 by GCC 3.2.1
using base arithmetic option LiDIA_ALL (LiDIA bigints and multiprecision floating point)
Using LiDIA multiprecision floating point with 15 decimal places.
Enter curve: [6, 0, 1, 0, 0]

Curve [6,0,1,0,0] :     Working with minimal curve [0,0,1,-24,45]
        [u,r,s,t] = [1,-3,-3,9]
No points of order 2
Basic pair: I=1152, J=-78192
disc=1306368
2-adic index bound = 2
By Lemma 5.1(a), 2-adic index = 1
2-adic index = 1
One (I,J) pair
Looking for quartics with I = 1152, J = -78192
Looking for Type 2 quartics:
Trying positive a from 1 up to 11 (square a first...)
(1,0,-18,4,69)  --trivial
Trying positive a from 1 up to 11 (...then non-square a)
Finished looking for Type 2 quartics.
Looking for Type 1 quartics:
Trying positive a from 1 up to 11 (square a first...)
Trying positive a from 1 up to 11 (...then non-square a)
(3,6,-12,-14,21)        --nontrivial...(x:y:z) = (1 : 2 : 1)
Point = [-3 : 9 : 1]
        height = 1.86324355221236
Rank of B=im(eps) increases to 1 (The previous point is on the egg)
Exiting search for Type 1 quartics after finding one which is globally soluble.
Mordell rank contribution from B=im(eps) = 1
Selmer  rank contribution from B=im(eps) = 1
Sha     rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer  rank contribution from A=ker(eps) = 0
Sha     rank contribution from A=ker(eps) = 0
Rank = 1
Points generating E(Q)/2E(Q):
Point [-6 : 27 : 1], height = 1.86324355221236
After descent, rank of points found is 1
Transferring points back to original curve [6,0,1,0,0]

Generator 1 is [-6 : 27 : 1]; height 1.86324355221236

The rank has been determined unconditionally.
The basis given is for a subgroup of full rank of the Mordell-Weil group
 (modulo torsion), possibly of index greater than 1.
Regulator (of this subgroup) = 1.86324355221236

 (4.4 seconds)

gp>  rpE([-6,27],6,10)
[-6, 27]
[1092/361, -140608/6859]
[-9584675/10614564, 146780171875/34582249512]
[386803581528/8108318895121, -29699040593262269943/23088527245364893831]
[-53170990338491386458/226537765418859862969, 1274069210009839548900881051368/3409658772817908353401089550547]
[-30738081772957847230689512975/210374216657690893470201900816, -8472649225243298104563809842697587183928411/96491435579663961818932360677510857604035136]
[-1377259259036510395864909512895515013842/6694733770364987932450821031704945314089, 25363677736885619177066591927452262345914680729841690255377/547772173577494249783411012790297759922899578543742309595563]
[-471391465492132899292284053502155365290392852319024/7152743465085223022592660505197272099659951296229729, -286631553684199588067347137538353215551646708752447709691402962406198849536/604935364190002832980816370706426345547827341444938815813187187751984366700433]
[-143007942924230104888508030771281319862869591997762647394940244291/269104638332386792617201605384303837916675946825678113638994802500, 9893338024692361033446469452812108036380148820664839778388966113675309524095131055643945407854577/139598828935790052655133508226706387668420819207903993666861474209568787229125209779197351410125000]
[373774068098231121705360118945601785081757811938220707114413840551296141326475916/302774047830944128617580741518671777487817830790080818448646965634032909844899289, 1149170838427624827043920631411923908654744247440486452756467615724068748539055404044103149822896201769765012245395457851/5268390652640659605597065580053061693591767267119437059709597516028797670684974457324532972428509290510034133190402304413]
time = 47 ms.

gp>  rpA([-6,27],6,10)
[6, 9/2]
[-1092/361, 2704/399]
[9584675/10614564, 27825625/5919786]
[-386803581528/8108318895121, 9590213595249/355665513944]
[53170990338491386458/226537765418859862969, 117524572543528506724/73821134521513832547]
[30738081772957847230689512975/210374216657690893470201900816, -41560377571055711514162812041/69156542956951756462642586100]
[1377259259036510395864909512895515013842/6694733770364987932450821031704945314089, 863259734523451025096188107541964659809/3835408391397917634802111513355883776438]
[471391465492132899292284053502155365290392852319024/7152743465085223022592660505197272099659951296229729, -43472638373709655638992488122412119323328255068416/6046583306971267026768895583514707948691252992698113]
[143007942924230104888508030771281319862869591997762647394940244291/269104638332386792617201605384303837916675946825678113638994802500, 46085244777980156541936323896560631378516743708855614441109717409/345572999084023473527570882471374701982351146438807662148359563350]
[-373774068098231121705360118945601785081757811938220707114413840551296141326475916/302774047830944128617580741518671777487817830790080818448646965634032909844899289, -109712572363773571003219263717695187294337456118391622125335575531194046894819401/620926638460683832954468995455734807229307121249323432272101362555075832637313172]
time = 22 ms.

gp>  rpC([-6,27],6,10)
[12, 9, 2]
[-22932, 51376, 7581]
[17415354475, 90655886250, 19286662788]
[-48313314547173312, 27308238704821075239, 1012761463276193384]
[260786531732120217365431085802, 1768882504220886840084123089612, 1111094560658606608142550260961]
[4634616260693472690729207646663793441556875, -19062319322092776616289097422144820713455236, 31719733604526868197482346507966751326795600]
[64559574486549980317349907710368345747664977687333438285188, 70633079277185536037357392627802552360212921466330995726803, 313818303038935967800629401307879557072745299086647462868546]
[33701981163120895164867564559185287734535708974701844385093792746728668164656, -3676650288839487655498193975692187238272222190933504323415323407784238463232, 511383092761914606382537209358159430359418185302179236640949683167744591530801]
[95266315755087087632468039211417799209402412087000230736655247847665190391340194047924785352104377, 23906857355163607622704087280211580510085409419566973805111759577975794441899658780869300176842450, 179267017777569988878568869022096177126826475571253573560504269179011099393693639090042834839967500]
[-13337982271442798739450103068878076232624792989913360165569478340606986072481237071691540631689503306619620432701020855856, -1909043046652405208837463331656426224461038789994915286201716526354220587884258693242649641845018093887004316671464378317, 10804374157815520167557367421158099952853295980774664308827029147440354069891253314323805944791861485903131867058801165924]
time = 17 ms.

[参考文献]

• [1]Joseph H. Silverman, John Tate(著), 足立 恒雄, 木田 雅成, 小松 啓一, 田谷 久雄(訳), "楕円曲線論入門", シュプリンガー・フェアラーク東京, 1995, ISBN4-431-70683-6, {3900円}.
  
• [2]Joseph H. Silverman, "The Arithmetic of Elliptic Curves", GTM 106, Springer-Verlag New York Inc., 1986, ISBN0-387-96203-4.
  
• [3]Dave Rusin, "For which values of n is a/b + b/c + c/a = n solvable?", 2003.08.01.
  
• [4]Henri Cohen, "Number Theory Volume I: Tools and Diophantine Equations", GTM 239, Springer-Verlag New York Inc., 2007, ISBN978-0-387-49922-2, p609.
  

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