又一道整数解问题

wayne

发现n=84的时候, PARI/Gp的ellrank计算秩是0<r<2. 没有找到生成元., 也就是 $x y+y^2=x^3+7829 x^2+2633925 x+226532601$
换成mwrank 的2-descent 也找不到.
---

1. zuse@konrazes-iMac progs % mwrank -v 10 -p 100 -o -x 100 -l -d -1
   
2. mwrank: unrecognized option `-1'
   
3. usage: mwrank [q p<precision> v<verbosity> b<hlim_q> x<naux>  c<hlim_c> l t o s d>]
   
4. Program mwrank: uses 2-descent (via 2-isogeny if possible) to
   
5. determine the rank of an elliptic curve E over Q, and list a
   
6. set of points that generate E(Q) modulo 2E(Q).
   
7. and finally saturate to obtain generating points on the curve.
   
8. For more details see the mwrank documentation.
   
9. For details of algorithms see the author's book.
   
10. 
    
11. Please acknowledge use of this program in published work,
    
12. and send problems to john.cremona@gmail.com.
    
13. 
    
14. eclib version 20250530, using NTL bigints and NTL real and complex multiprecision floating point
    
15. Using multiprecision floating point with 100 bits precision.
    
16. Enter curve: 1 7829 0 2633925 226532601
    
17. Curve [1,7829,0,2633925,226532601] :        Working with minimal curve [1,-1,0,-17798460,28906041276] via [u,r,s,t] = [1,-2610,0,1305]
    
18. 
    
19. 1 points of order 2:
    
20. [2436:-1218:1]
    
21. 
    
22. (c,d)  =(29229,2784)
    
23. (c',d')=(-58458,854323305)
    
24. Using 2-isogenous curve [0,-58458,0,854323305,0] (minimal model [1,-1,0,-17799330,28903074750])
    
25. -------------------------------------------------------
    
26. First step, determining 1st descent Selmer groups
    
27. -------------------------------------------------------
    
28. Finding els gens for E (c= 29229, d= 2784)
    
29. Support (length 4): [ -1 2 3 29 ]
    
30. Adding (torsion) els generator #1: d1 = 174
    
31. Adding els generator #2: d1 = -6
    
32. After els sieving, nelsgens = 2
    
33. 2-rank of S^{phi}(E') = 2
    
34. (els)gens: [ 174 -6 ]
    
35. 
    
36. Finding els gens for E' (c'= -58458, d'= 854323305)
    
37. Support (length 5): [ -1 3 5 11 173 ]
    
38. Adding (torsion) els generator #1: d1 = 28545
    
39. Adding els generator #2: d1 = 33
    
40. After els sieving, nelsgens = 2
    
41. 2-rank of S^{phi'}(E) = 2
    
42. (els)gens: [ 28545 33 ]
    
43. After first local descent, rank bound = 2
    
44. rk(S^{phi}(E'))=        2
    
45. rk(S^{phi'}(E))=        2
    
46. 
    
47. -------------------------------------------------------
    
48. Second step, determining 2nd descent Selmer groups
    
49. -------------------------------------------------------
    
50. ...skipping -- no second descent requested
    
51. No second local descent done, rank bound = 2
    
52. rk(phi'(S^{2}(E)))<=        2
    
53. rk(phi(S^{2}(E')))<=        2
    
54. rk(S^{2}(E))<=        3
    
55. rk(S^{2}(E'))<=        3
    
56. 
    
57. Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
    
58. -------------------------------------------------------
    
59. 1. E(Q)/phi(E'(Q))
    
60. -------------------------------------------------------
    
61. (c,d)  =(29229,2784)
    
62. (c',d')=(-58458,854323305)
    
63. Adding (torsion) gls generator #1: d1 = 174
    
64. (-6,0,29229,0,-464):  no rational point found (hlim=10)
    
65. (-29,0,29229,0,-96):  no rational point found (hlim=10)
    
66. After first global descent, this component of the rank
    
67.         has lower bound 0
    
68.         and upper bound 1
    
69.         (difference =   1)
    
70. 
    
71. After gls sieving, nglsgens = 1
    
72. shortfall in rank from first  descent = 1
    
73. (gls)gens: [ 174 ]
    
74. -------------------------------------------------------
    
75. 2. E'(Q)/phi'(E(Q))
    
76. -------------------------------------------------------
    
77. Adding (torsion) gls generator #1: d1 = 28545
    
78. (33,0,-58458,0,25888585):  no rational point found (hlim=10)
    
79. (865,0,-58458,0,987657):  no rational point found (hlim=10)
    
80. After first global descent, this component of the rank
    
81.         has lower bound 0
    
82.         and upper bound 1
    
83.         (difference =   1)
    
84. 
    
85. After gls sieving, nglsgens = 1
    
86. shortfall in rank from first  descent = 1
    
87. (gls)gens: [ 28545 ]
    
88. 
    
89. -------------------------------------------------------
    
90. Summary of results:
    
91. -------------------------------------------------------
    
92.         0 <= rank(E) <= 2
    
93.         #E(Q)/2E(Q) >= 2
    
94. 
    
95. Information on III(E/Q):
    
96.         #III(E/Q)[phi'] <= 2
    
97.         #III(E/Q)[2]    <= 4
    
98. Information on III(E'/Q):
    
99.         #III(E'/Q)[phi] <= 2
    
100.         #III(E'/Q)[2]   <= 4
     
101. 
     
102. 
     
103. Used descent via 2-isogeny with isogenous curve E' = [1,-1,0,-17799330,28903074750]
     
104. 0 <= rank <= 2
     
105. Searching for points (bound = 8)...P1 = [0:1:0]         is torsion point, order 1
     
106. P1 = [2436:-1218:1]         is torsion point, order 2
     
107. P1 = [2436:-1218:1]         is torsion point, order 2
     
108. P1 = [2437:-1132:1]         is torsion point, order 3
     
109. P1 = [2610:13746:1]         is torsion point, order 6
     
110. done:
     
111.   found points which generate a subgroup of rank 0
     
112.   and regulator 1
     
113. Processing points found during 2-descent...Processing 0 points ...
     
114. Finished processing the points (which had rank 0)
     
115. done:
     
116.   now regulator = 1
     
117. 
     
118. 
     
119. Regulator = 1
     
120. 
     
121. The rank has not been completely determined,
     
122. only a lower bound of 0 and an upper bound of 2.
     
123. 
     
124. [[0,2],[]]
     
125. 
     
126. (0.344131 seconds)
     

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wayne

最近几天疯狂的计算到了n=3000，跳过了几个特别难计算的. 主要是这个曲线很奇怪,比如n=38,秩为1,但是最小解是一个1584369位数.见链接https://nestwhile.com/res/abcn/38.txt

对于其他的数 我也一一比对了一下这个日本人的 答案.https://www.kaynet.or.jp/~kay/misc/de95.html , 高度一致.
但我也意外捡到漏了, 上面的文件是我第一次计算，n=34(最小的秩为2)计算漏掉了更小的解，第二次因为重构了代码，可以非常严谨的枚举所有的系数组合$a*P_1+b*P_2+c*T_i$，
最小解分别是268,268,270位数(那个日本人网站给的是300,300,302位数，也漏掉了,是我下面的第二组解).

1. {268,268,270,(1889807555411091146590400229041950860706846754777218000981139464741944939021464140413296047608199984680425804169132554677689744493791660802796287444463080577716192263907985576632296781215724785871489647121514411873306282368019554698109966469732018135083199775597913437, 2794291547909379203432499959740262148947817104665018180412543011022527575937517870792378258966371376798337658373171175801391436403242247494885661645741348494133328303809904629592092936992287395570654705306800862955159283484821880946612803103265494437171360632024214733, 159123409641885803304570756882964382534999874891625661949254586642561013606721641991654177332112386388235686383599629915455542793766219475821358798060398824201885575403639653767532771606465462096742849874914001403201998808358992284469525699773885501422890452911968468047),[(-4, 11, 2), (-4, 11, 5), (-4, 11, 6), (4, -11, 2), (4, -11, 5), (4, -11, 6)] }
   
2. {300,300,302,(446761075375415094348664370149205699961945113086925822371832900750539646807140361024789308922481959455472679127978761888342076841924631439213403725443526605407749656992653976113477849781516494847762257335499847143737766700178413417289056731320095586508223271369274841372262284541144085392037100432399, 490748366010305182990474150656636362190389764266925090417492096168051776765545482593353235835437018048897846423001695225704506529352474238680902919831391243154823510294626841155092057036559845199980906348743965722194847215049183296615581521690304391296029446989128920698451724057626849201514644099616, 31848123069527289200571887310242148475162436956073309844967234018722478242668230590248519712160838595089154336300240115970398964214358661950158616803336201395663130347314048559298184194270167293021176436056763442575705224732883855569939968854927948969578550229819747639695012262375261234732019528970915),[(0, -11, 0), (0, -11, 1), (0, -11, 3), (0, -11, 4), (0, 11, 0), (0, 11, 1), (0, 11, 3), (0, 11, 4)] }
   
3. {380,380,382,(17521087258566732239045041167993878571069231737745335563049099659579044258928819026367486913595910332843648281609653891387248209943022229637983266823945089097971258756895959655607374620790848783305784206860914263883811477561807221891518642790957188822964761314072110856109343853759977091843097927716457819020919065655517791974009618969966580262885938173324935123634700022865785075, 44293341628965071156752424368919423342932977379680792445642681061293819101775066554256517248955472985292840639995101480064239772231520040534816943325120397775547440458015980768454660568644137922808043647835458004715128247830038021194120813111492307771604681002309794820051005722456457598325105739222270633637683073449148809695775069102243477101056257501316223960599920930999332896, 2099892397440982457043716450270178469160804552368229908901651370782881940105558740604780301824455819105304731841225090647127108722379011324434121430687444934563776742870832558500661312445092169735243193205881998693749849274908414528601691779923362492765734493244388371182541547685155370446144001768353088740401476820863116337019728073087222050453302321725674757788917094487317647679),[(-8, 11, 0), (-8, 11, 1), (-8, 11, 3), (-8, 11, 4), (8, -11, 0), (8, -11, 1), (8, -11, 3), (8, -11, 4)] }

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wayne

统计一下情况.
1) 秩的计算跟生成元个数不一致的情况.
1.1) 其中计算出至少一个生成元的有

1. 658,934,1120,2004,2086,2150,2228,2554,2588,2614,2656,2744,2756,2974,3030,3084,3106,3146,3196,3222,3376,3388,3426,3628,3682,3786,3900,3906,3920,4044,4198,4308,4498,4548,4720,4822,4842,4856,4918,5008,5048,5070,5144,5158,5162,5216,5224,5242,5260,5446,5496,5516,5542,5554,5558,5722,5758,5924,6048,6212,6296,6298,6414,6444,6538,6730,6870,6872,6946,6966,7070,7082,7086,7104,7354,7434,7624,7724,7836,7850,8074,8114,8236,8268,8440,8454,8460,8482,8534,8560,8566,8576,8580,8626,8678,8716,8746,8828,8934,9018,9106,9126,9244,9316,9398,9400,9466,9478,9594,9618,9628,9722,9744,9764,9778,9802,9880,9890,9918,9970

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1.2) 0个生成元的有

1. {84,122,156,198,220,224,236,248,274,304,318,328,374,378,390,402,404,408,428,440,444,460,480,484,488,500,526,536,548,562,572,584,616,626,636,638,644,658,670,678,688,692,698,704,718,728,730,744,746,758,766,768,780,784,796,798,808,818,822,832,836,838,840,846,850,854,880,882,892,896,908,916,918,920,926,934,944,948,950,954,966,972,974,976,980,1000,1004,1008,1018,1022,1034,1038,1046,1050,1056,1058,1062,1074,1086,1088,1092,1098,1106,1112,1116,1118,1120,1124,1140,1148,1150,1164,1174,1176,1178,1186,1190,1198,1204,1210,1216,1220,1224,1228,1234,1242,1244,1250,1254,1256,1260,1276,1286,1288,1290,1292,1294,1298,1300,1302,1306,1312,1316,1326,1328,1332,1342,1344,1354,1372,1376,1390,1396,1404,1406,1416,1424,1426,1428,1430,1432,1434,1444,1448,1454,1456,1466,1468,1472,1478,1484,1488,1490,1494,1496,1500,1506,1526,1538,1540,1548,1552,1554,1578,1580,1582,2650,....,9990,9992,10000}

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2) 然后秩为1,且成功计算出1个生成元的

1. {4,6,10,12,14,16,18,24,28,32,38,40,42,44,46,48,50,58,60,66,68,76,80,82,92,102,104,106,110,112,116,124,126,128,130,132,136,146,158,162,170,176,178,180,182,184,186,196,200,206,216,218,226,230,232,234,238,254,256,266,270,276,282,290,310,312,314,320,330,332,334,336,338,342,346,348,358,360,362,364,370,372,380,382,386,388,394,398,410,420,422,438,442,450,452,454,458,468,490,492,498,510,512,522,524,530,534,538,540,544,554,564,568,578,598,600,602,606,612,622,628,630,634,642,650,654,660,664,684,686,690,710,714,724,726,732,736,750,756,762,774,776,778,794,802,806,810,812,820,824,828,848,852,866,876,884,886,890,894,898,900,902,904,906,912,924,928,932,938,942,946,952,960,978,984,992,996,998,1016,1024,1030,1040,1042,1052,1054,1060,1066,1068,1070,1084,1094,1096,1100,1110,1114,1126,1130,1144,1156,1158,1166,1168,1170,1182,1184,1194,1206,1230,1232,1238,1246,1258,1264,1268,1272,1280,1308,1320,1322,1334,1348,1352,1360,1362,1366,1368,1370,1380,1382,1384,1392,1394,1400,1408,1418,1422,1442,1450,1460,1492,1498,1502,1516,1518,1522,1530,1542,1544,1558,1560,1568,1570,1574,1586,1588,1590,1612,1614,1626,1632,1640,1648,1670,1672,1682,1684,1700,1710,1724,1726,1740,1746,1750,1752,1768,1772,1778,1786,1788,1808,1810,1818,1822,1824,1846,1850,1860,1878,1894,1896,1900,1908,1912,1918,1928,1948,1952,1972,1980,1990,1992,2010,2028,2052,2054,2074,2078,2084,2090,2104,2106,2116,2118,2122,2142,2148,2160,2162,2182,2202,2206,2214,2216,2218,2232,2234,2252,2258,2260,2272,2306,2320,2322,2330,2332,2340,2342,2346,2350,2360,2366,2380,2396,2400,2404,2406,2422,2438,2460,2470,2476,2488,2496,2498,2500,2510,2512,2542,2550,2558,2562,2564,2570,2578,2580,2582,2610,2612,2640,2642,2648,2652,2654,2666,2698,2700,2714,2724,2726,2734,2740,2746,2760,2776,2822,2830,2832,2834,2844,2860,2868,2888,2890,2896,2900,2904,2912,2918,2930,2964,2980,2984,2994,3012,3046,3050,3062,3066,3098,3132,3134,3138,3162,3198,3202,3206,3246,3260,3280,3332,3336,3346,3354,3360,3362,3364,3372,3410,3414,3452,3454,3482,3506,3512,3522,3570,3586,3592,3602,3614,3626,3644,3658,3680,3684,3712,3716,3736,3780,3782,3790,3796,3816,3834,3858,3864,3880,3882,3926,3934,3978,3980,3984,3988,4008,4020,4034,4036,4062,4080,4084,4120,4124,4130,4140,4154,4178,4194,4218,4224,4260,4264,4272,4282,4318,4324,4326,4344,4362,4376,4386,4412,4414,4418,4420,4426,4428,4436,4438,4440,4460,4472,4494,4510,4550,4560,4590,4592,4596,4620,4632,4658,4704,4712,4714,4722,4744,4750,4754,4794,4844,4862,4884,4902,4926,4936,4952,4970,5000,5002,5004,5006,5014,5042,5044,5102,5110,5116,5120,5122,5136,5146,5200,5202,5226,5236,5262,5270,5276,5288,5294,5312,5314,5378,5388,5420,5424,5436,5448,5456,5464,5498,5510,5522,5570,5614,5626,5640,5674,5676,5682,5690,5712,5732,5742,5748,5762,5792,5794,5804,5808,5810,5814,5826,5828,5856,5860,5868,5874,5882,5910,5914,5916,5932,5938,5956,5960,5982,5990,5996,6004,6010,6032,6042,6058,6082,6094,6102,6132,6166,6180,6200,6234,6242,6248,6266,6290,6306,6324,6334,6348,6364,6378,6386,6454,6484,6500,6512,6524,6610,6612,6626,6636,6638,6646,6648,6666,6670,6680,6682,6738,6742,6744,6756,6770,6792,6808,6810,6812,6820,6842,6856,6860,6864,6866,6920,6926,6948,6950,6958,6960,6970,6972,7004,7008,7010,7026,7032,7050,7054,7078,7092,7098,7144,7180,7190,7200,7204,7236,7250,7260,7264,7286,7300,7314,7326,7342,7372,7384,7426,7444,7472,7494,7502,7516,7544,7558,7582,7588,7626,7660,7662,7664,7670,7682,7692,7714,7730,7750,7760,7762,7772,7774,7776,7820,7826,7830,7862,7866,7904,7916,7918,7954,7964,7970,8014,8030,8034,8052,8066,8082,8132,8134,8148,8156,8160,8170,8224,8246,8248,8272,8290,8318,8320,8322,8354,8356,8432,8502,8524,8532,8542,8604,8632,8692,8714,8726,8752,8784,8786,8810,8812,8818,8824,8832,8874,8880,8884,8908,8940,8952,8980,9002,9004,9038,9044,9074,9078,9094,9122,9124,9130,9180,9210,9214,9290,9294,9384,9386,9390,9408,9482,9492,9510,9522,9526,9528,9542,9560,9568,9578,9584,9606,9612,9644,9670,9692,9696,9710,9746,9748,9758,9762,9780,9784,9830,9832,9844,9872,9888,9902,9904,9906,9916,9930}

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3)秩为2,且成功计算出2个生成元的

1. 34,94,98,114,144,152,160,166,214,228,244,258,260,268,300,324,384,406,426,466,486,506,520,556,580,582,592,604,618,620,624,676,706,722,754,760,782,790,816,826,860,874,940,970,982,994,1006,1014,1020,1048,1104,1136,1160,1218,1248,1356,1410,1412,1420,1486,1524,1564,1576,1592,1594,1636,1644,1674,1688,1734,1748,1764,1844,1854,1930,1984,2040,2058,2082,2156,2166,2302,2356,2382,2414,2416,2442,2448,2454,2504,2522,2536,2680,2706,2732,2780,2792,2800,2806,2818,2846,3000,3032,3130,3204,3214,3254,3270,3272,3290,3298,3322,3380,3442,3462,3480,3532,3544,3546,3640,3664,3714,3730,3754,3820,3830,3844,3848,3910,4030,4090,4106,4160,4164,4170,4234,4246,4294,4382,4434,4450,4506,4512,4638,4798,4834,4960,5028,5130,5156,5174,5184,5274,5318,5360,5366,5382,5406,5434,5440,5580,5620,5644,5650,5704,5806,5822,5854,5888,5922,5986,6060,6084,6092,6158,6162,6302,6368,6530,6694,6702,6748,6790,6916,6928,6990,7062,7120,7208,7358,7474,7484,7486,7706,7736,7758,7868,7910,7934,7960,7982,7984,7996,8084,8106,8198,8278,8324,8408,8490,8506,8730,8770,8794,9006,9062,9088,9108,9254,9258,9336,9344,9360,9452,9630,9662,9664,9706,9940,9962,9994

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4) 秩为3,且成功计算出3个生成元的

1. {424,680,1336,1766,2390,2866,3310,3578,3678,3932,4002,4534,4696,4974,6976,8194,8376,8538,9720}

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wayne

我们重点考虑一下n=156,因为 ellrank 能够确定 秩为1, 但是找不到生成元

1. k=156;n=2*k+6;aaa=[n-3,0,n-1,0,0];E=ellinit(aaa);
   
2. ellrank(E,1)
   
3. %33 = [1, 1, 0, []]

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然后,我们尝试一个命令ellheegner, 可以找到 一个非扰子群里的点 ,运气很不错,找到了!

1. (20251918.13:19:45)> ellheegner(E)
   
2.   *** ellheegner: Warning: increasing stack size to 16000000.
   
3.   *** ellheegner: Warning: increasing stack size to 32000000.
   
4. %26 = [-21567652354577128476907655095/1928481565707324432272209, 8212016982999099943163620113592958136869291/2678078097905270420857799229762361177]

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代入得到第一个解

1. 1(20255818.13:58:16)> P=%26
   
2. %73 = [-21567652354577128476907655095/1928481565707324432272209, 8212016982999099943163620113592958136869291/2678078097905270420857799229762361177]
   
3. (20255818.13:58:33)> [X,Y]=P;[a,b,c]=[-Y*(1+X)/(X^2+Y),-X*(1+X)/(X-Y),1];Q=[a+b-c,a+c-b,-a+b+c]/2;Q/gcd(Q)
   
4. %74 = [1577133783148999215225003543774692512573807, 1170188861606405478100419242093170984961135, -1159922227598355442255065853304348643551587]
   
5. (20255818.13:58:35)>

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也可以这么玩.

1. (20251818.14:18:40)> k=156;n=2*k+6;aaa=[n-3,0,n-1,0,0];E=ellinit(aaa);
   
2. (20251818.14:18:53)> C=ell2cover(E); #C
   
3. %2 = 2
   
4. (20251918.14:19:05)> [Q,M] = C[1]; Q
   
5. %3 = x^4 - 634*x^3 + 99219*x^2 - 634*x + 1
   
6. (20251918.14:19:11)> M
   
7. %4 = [8586/y^2*x^4 - 5241912/y^2*x^3 + 820342056/y^2*x^2 - 5241912/y^2*x + ((-8269*y^2 + 8586)/y^2), -50403/y^3*x^6 + 15776139/y^3*x^5 + ((-1352295*y - 252015)/y^3)*x^4 + 825601140/y^2*x^3 + ((-129203873820*y + 252015)/y^3)*x^2 + ((825601140*y - 15776139)/y^3)*x + ((1302209*y^3 - 1352295*y + 50403)/y^3)]
   
8. (20251918.14:19:17)> P=substvec(M,[x,y],[45200310930197293707183074/12095477219811988059529574421,18686816920613387493105704786157018311717222175520820107/146300569174990740113939291140759834091688126745377485241])
   
9. %5 = [6805887082526065838682175257968959492106933949096834708487368287119878017519983577048885444442414482220485963910101/349197126624522805969744379309666404164863525844818481815962019784843735708648321290130776639123799033855491449, -46799983661138854628257420706546915366246795905534173853359117558335947453402112408367011005128762239849648022135158939381732538673885802046207610284419466388179166154964313/6525382774436708407621227679230369448415366527672062331744308450019600397212540261194944531619957017058925978021075361741423565454988646369903214250804795254005765043]
   
10. (20252018.14:20:06)> ellisoncurve(E,P)
    
11. %6 = 1
    
12. (20252018.14:20:23)> [X,Y]=P;[a,b,c]=[-Y*(1+X)/(X^2+Y),-X*(1+X)/(X-Y),1];Q=[a+b-c,a+c-b,-a+b+c]/2;Q/gcd(Q)
    
13. %7 = [398717662788278063072034394210418461757935392331603882543448093388903194940038490461818377605135414838988539013550515861716300061767904210940334832437394742355375008179, 532321442563236465779991317198238652228380951479418520844588486903356624982119373381110397693722819524196031217440866557051991527166674875024752244479452957765341974861, -529839212942895697712495031334117208606168799417368319450843526239186475649569842050726705079333481183550907084304604275406580916430517181578103943098525173290016853069]

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wayne

但这些点都不是 生成元. 还是magma厉害,吊打PARI/Gp和Sagemath, 瞬间给出结果, https://magma.maths.usyd.edu.au/calc/

1. SetClassGroupBounds("GRH");
   
2. k:=156;
   
3. E := EllipticCurve([0, 4*k^2 + 12*k - 3, 0, 32*(k+3), 0]);
   
4. print E;
   
5. gens := Generators(E);
   
6. print gens;

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1. Elliptic Curve defined by y^2 = x^3 + 99213*x^2 + 5088*x over Rational Field
   
2. [ (1272 : -403224 : 1), (-25426032308736047814660503136/40703637875493610528301553961 : 1548018481548026576021918683596507891038604288/8212016982999099943163620113592958136869291 : 1) ]
   

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之前说的84计算很困难, magma直接 给出来是秩为0,只有有限阶的生成元 [ (696 : -120408 : 1) ]

wayne

不是所有的n都能保证$\frac{a}{b + c} +\frac{b}{c + a} + \frac{c}{a + b} = n$有整数解的.然后这其中又只有一部分n只能给出负正整数的解.然后我们重点关注可以拿到正整数解的那些n.

下面就是我统计的n<=1000以内可以保证得到a,b,c的正整数解的 全部 秩和生成元,对应的椭圆曲线方程是$y^2=(4 n^2+12 n-3) x^2+32 (n+3) x+x^3$
(得益于神秘而又给力的magma, 多处卡壳的地方用 magma来补充, 原则上我可以整理到10000,但手工的地方太多,越到后面,magma的占比越大.在线使用magma又特别费劲.)
同时我也把OEIS的两个数列扩充了一倍.https://oeis.org/A369896, https://oeis.org/A283564
格式是n, 秩,有限阶的生成元,无限阶的生成元. 不难发现,扰子群的生成元是$[8 n+24,-8 (n+3) (2 n+5)]$, 有些生成元的分子和分母 相当的恐怖,这难怪PARI/Gp和Sagemath都失效了.

1. {4, 1, {{56, -728}, {-4, -28}}}
   
2. {6, 1, {{72, -1224}, {-32/9, -1088/27}}}
   
3. {10, 1, {{104, -2600}, {-36, -780}}}
   
4. {12, 1, {{120, -3480}, {-20/9, -1340/27}}}
   
5. {14, 1, {{136, -4488}, {-68/49, -11220/343}}}
   
6. {16, 1, {{152, -5624}, {-152/169, 45752/2197}}}
   
7. {18, 1, {{168, -6888}, {-6069/625, -5737704/15625}}}
   
8. {24, 1, {{216, -11448}, {-24/49, 4824/343}}}
   
9. {28, 1, {{248, -15128}, {-49/169, 4382/2197}}}
   
10. {32, 1, {{280, -19320}, {-56/81, 26936/729}}}
    
11. {34, 2, {{296, -21608}, {-1, -62}, {196/81, 131068/729}}}
    
12. {38, 1, {{328, -26568}, {-41/121, -21894/1331}}}
    
13. {42, 1, {{360, -32040}, {-3528/361, 5766264/6859}}}
    
14. {46, 1, {{392, -38024}, {-1089/625, 2451966/15625}}}
    
15. {48, 1, {{408, -41208}, {-833/9, -245854/27}}}
    
16. {58, 1, {{488, -59048}, {-800/1681, -3287680/68921}}}
    
17. {60, 1, {{504, -63000}, {-224, 27328}}}
    
18. {66, 1, {{552, -75624}, {-760013311246836/6643274277025, -263392287178690286463012/17122740501689471375}}}
    
19. {76, 1, {{632, -99224}, {-632/5329, 2395912/389017}}}
    
20. {82, 1, {{680, -114920}, {-68/529, -128180/12167}}}
    
21. {92, 1, {{760, -143640}, {-152/529, -545832/12167}}}
    
22. {94, 2, {{776, -149768}, {169/784, -1067261/21952}, {-164836/900601, -21849629732/854670349}}}
    
23. {98, 2, {{808, -162408}, {-2213408/332929, 252525143424/192100033}, {121/784, -833547/21952}}}
    
24. {102, 1, {{840, -175560}, {-32/9, 19648/27}}}
    
25. {112, 1, {{920, -210680}, {-16629134116/232656699025, -44213063270017852/112220795491213625}}}
    
26. {114, 2, {{936, -218088}, {-8/9, 5320/27}, {-104/1369, -245336/50653}}}
    
27. {116, 1, {{952, -225624}, {-55693700/670761, -10705405659220/549353259}}}
    
28. {126, 1, {{1032, -265224}, {-1418543062831986302475272/260393118777452893327489, 183481874964426384170903625944484403352/132875299028344770469380547125612737}}}
    
29. {130, 1, {{1064, -281960}, {-49284/187489, -4911246060/81182737}}}
    
30. {132, 1, {{1080, -290520}, {-1830125/6561, 39494956600/531441}}}
    
31. {136, 1, {{1112, -308024}, {-1112/17689, 10307128/2352637}}}
    
32. {144, 2, {{1176, -344568}, {-21/361, -24612/6859}, {-33286701/355586449, -116449528092576/6705293668793}}}
    
33. {146, 1, {{1192, -354024}, {-200, -58920}}}
    
34. {152, 2, {{1240, -383160}, {4/81, 15884/729}, {-44180/27889, -2226956820/4657463}}}
    
35. {156, 1, {{1272, -403224}, {-25426032308736047814660503136/40703637875493610528301553961, 1548018481548026576021918683596507891038604288/8212016982999099943163620113592958136869291}}}
    
36. {158, 1, {{1288, -413448}, {-161/81, 456274/729}}}
    
37. {160, 2, {{1304, -423800}, {49/256, 284375/4096}, {-121, -39050}}}
    
38. {162, 1, {{1320, -434280}, {-5/81, -6580/729}}}
    
39. {166, 2, {{1352, -455624}, {-396258733/324396121, -2343121976373256/5842698535331}, {-416/5625, 6168448/421875}}}
    
40. {178, 1, {{1448, -522728}, {-2612111675168/14356725606729, 3083067790302174769664/54398020955487562683}}}
    
41. {182, 1, {{1480, -546120}, {-392, -143640}}}
    
42. {184, 1, {{1496, -558008}, {-24147396/3073009, 15659932222116/5386984777}}}
    
43. {186, 1, {{1512, -570024}, {-168/3721, -836136/226981}}}
    
44. {196, 1, {{1592, -632024}, {-9604/235225, -31371956/114084125}}}
    
45. {198, 1, {{1608, -644808}, {-342305015706874574796614242565866568/3683074469296260781350290648449, 169086090685187087743189719612573762104451666045652184/7068312631889204246886009531685585716276411457}}}
    
46. {200, 1, {{1624, -657720}, {-224/529, -1975680/12167}}}
    
47. {206, 1, {{1672, -697224}, {-32039892829845581001156/8522095529020193750161, -1221061980595589195474304207046120212/786718924469665152815830002907591}}}
    
48. {214, 2, {{1736, -751688}, {-34225/12769, 1655129510/1442897}, {-217/361, 1720810/6859}}}
    
49. {218, 1, {{1768, -779688}, {-416/5041, 9662016/357911}}}
    
50. {228, 2, {{1848, -851928}, {14193520000/1570009, 8336165321844800/1967221277}, {-1203125/4721929, 1114286327000/10260751717}}}
    
51. {232, 1, {{1880, -881720}, {-2590264/100489, 383170294808/31855013}}}
    
52. {244, 2, {{1976, -974168}, {4/49, -16276/343}, {-4/25, -8756/125}}}
    
53. {258, 2, {{2088, -1087848}, {1197580288/970260201, 19601767519293952/30222635000949}, {-288/1849, 5751936/79507}}}
    
54. {266, 1, {{2152, -1155624}, {-941549080041831436064244/43982533790005519051249, 105561659929775336815028374541489695236/9224022794641422162186769630789193}}}
    
55. {268, 2, {{2168, -1172888}, {-7700764516/225819792025, -696336486698736748/107310694269240125}, {-361, 194446}}}
    
56. {270, 1, {{2184, -1190280}, {-87066341024/207313570489, -20752356253790112640/94393392974340013}}}
    
57. {276, 1, {{2232, -1243224}, {-16380152/10609, -933994546168/1092727}}}
    
58. {282, 1, {{2280, -1297320}, {-7853460/37442161, -25337337449700/229108583159}}}
    
59. {300, 2, {{2424, -1466520}, {4/441, -100540/9261}, {-24/841, 107448/24389}}}
    
60. {304, 1, {{2456, -1505528}, {-38373884614460440506499741521/15418859847919486053458571961, 2895918515507626816007488805900862823371686058/1914601484477274696722566114082446750362541}}}
    
61. {310, 1, {{2504, -1565000}, {-484/169, -3902140/2197}}}
    
62. {312, 1, {{2520, -1585080}, {-115320/169, -939125160/2197}}}
    
63. {314, 1, {{2536, -1605288}, {-6757172/100821681, -33706118428100/1012350498921}}}
    
64. {318, 1, {{2568, -1646088}, {-8760908019032890502329684578218514722888/16900516085362451620946799104403513529, 22998905430310632564634203183296596566628845614735528162361544/69478422628359179067203671447552037893967170788684835667}}}
    
65. {332, 1, {{2680, -1792920}, {-536/25, 1786488/125}}}
    
66. {336, 1, {{2712, -1836024}, {-28355912753/1718185401, 792784796438368286/71220503056851}}}
    
67. {338, 1, {{2728, -1857768}, {-114097732849800/1082184558961, 80573712433655657451480/1125776035180508041}}}
    
68. {346, 1, {{2792, -1946024}, {-95481/32761, 11963108658/5929741}}}
    
69. {348, 1, {{2808, -1968408}, {-1176, -821016}}}
    
70. {362, 1, {{2920, -2128680}, {-7626125/13564489, 20013678336600/49958012987}}}
    
71. {364, 1, {{2936, -2152088}, {-26359774662762101792196/274069049372442025, -9134668494200304825052561045079508/143479544784370580919791125}}}
    
72. {378, 1, {{3048, -2319528}, {-45746672325503868658010097964819816918168455200/244688556238850422158885678557904788096168121, -543010858736847355816156929843375259756368101443940045403499607822188160/3827546606829998582024722009584767419631258934868624644231609070419}}}
    
73. {382, 1, {{3080, -2368520}, {-135495976/11881, 11217405903704/1295029}}}
    
74. {384, 2, {{3096, -2393208}, {-3865184/114511401, -19729655679808/1225386502101}, {345885604/505215529, 6084576615107164/11355729445333}}}
    
75. {388, 1, {{3128, -2442968}, {-294847198970432089464/3445468036920897331969, -11746050495012234850661891650673352/202242459465630351525966411527297}}}
    
76. {402, 1, {{3240, -2621160}, {-7237017742239021874848487112/103774824774698382610110217129, 1590441733022262351393841869048082101836534408/33430125971694557938357460037437103714629867}}}
    
77. {406, 2, {{3272, -2673224}, {5795999558328627076/85700658226225, -45901794384575889577421813924/793370844006240019625}, {-968/81, -7094296/729}}}
    
78. {408, 1, {{3288, -2699448}, {-7153415871587013564472505461540056/84347724033560118146846384965181281, -1491888102915124990050734927509683330754675804411924056/24496839269817973994348500368677272193337939307725679}}}
    
79. {422, 1, {{3400, -2886600}, {-56411712425/2489311449, -2382868548186681950/124199216124957}}}
    
80. {424, 3, {{3416, -2913848}, {-324, -275652}, {-6100, -5169140}, {-4941/5329, 303807816/389017}}}
    
81. {426, 2, {{3432, -2941224}, {-596062500/4012009, 1020619137169500/8036054027}, {-143818389/1555009, -153310481165556/1939096223}}}
    
82. {438, 1, {{3528, -3108168}, {-36/25, 157212/125}}}
    
83. {466, 2, {{3752, -3515624}, {906187592/121466887441, -536523155698626296/42333761077824761}, {-715716/235225, 323639616348/114084125}}}
    
84. {486, 2, {{3912, -3822024}, {-5939210254853718152/35100138664369, 31104361754914602045177144232/207952070222184722153}, {529870438084/49491225, 3654836595429895252/348170767875}}}
    
85. {492, 1, {{3960, -3916440}, {-6716405/1929321, -9186206137640/2679826869}}}
    
86. {506, 2, {{4072, -4141224}, {-98678142752/33465945969, -18273340299636533888/6122159757730953}, {-200/121, -2222280/1331}}}
    
87. {510, 1, {{4104, -4206600}, {-8649/2809, 467740710/148877}}}
    
88. {512, 1, {{4120, -4239480}, {-2578593464/95124746929, 531581890850542392/29338659822082967}}}
    
89. {522, 1, {{4200, -4405800}, {-7548855188819989368200/162595790175012489, -3118765494736481108374345713628600/65563774120985150485488987}}}
    
90. {536, 1, {{4312, -4644024}, {-139055225571757344072357176/1525834634056682374018594225, -5339602419580479437567739330722996615497032/59602050720379771514579682825744883268375}}}
    
91. {548, 1, {{4408, -4853208}, {-17044326944843180846993458106974676/31156006611408113192729613599038225, 3261916355539217456330108408822455856824939268144305932/5499366717852972876444750884947066420595185713541625}}}
    
92. {556, 2, {{4472, -4995224}, {-584758299906612/194216471928649, -9064658776803390595438812/2706631244783745261893}, {-1036324/332929, -665177445244/192100033}}}
    
93. {564, 1, {{4536, -5139288}, {-29053268376/79477225, 292891263584809032/708539460875}}}
    
94. {578, 1, {{4648, -5396328}, {-1725859047200/6446521, 4544469804463752960/16367716819}}}
    
95. {580, 2, {{4664, -5433560}, {-676, -785980}, {-49615939268984/277855561, -896113197173250263768/4631574346309}}}
    
96. {582, 2, {{4680, -5470920}, {-3744/49, -30580992/343}, {86436/10609, 10398439548/1092727}}}
    
97. {592, 2, {{4760, -5659640}, {2626492792/16719921, -12749262983263576/68367756969}, {-4/289, 12412/4913}}}
    
98. {604, 2, {{4856, -5890328}, {-29476031090222500/172484058174624001, 14238612167091717694848127100/71634699341103501277936001}, {-1616703224/426409, -1276800261620776/278445077}}}
    
99. {606, 1, {{4872, -5929224}, {-4901/81, 53585272/729}}}
    
100. {612, 1, {{4920, -6046680}, {-446520/786769, 480338378040/697864103}}}
     
101. {618, 2, {{4968, -6165288}, {-276/5041, -21215844/357911}, {-328517556/1504586521, 15313175832929076/58361406563069}}}
     
102. {620, 2, {{4984, -6205080}, {-3204/361, -75613332/6859}, {-162030456/169, 1612840395096/2197}}}
     
103. {622, 1, {{5000, -6245000}, {-4333750185000625/3429729094804321, -314873968761591703988131250/200858274008190731764369}}}
     
104. {624, 2, {{5016, -6285048}, {-5217476/16966161, 26318486910628/69883617159}, {2653464/17161, -434890174488/2248091}}}
     
105. {642, 1, {{5160, -6651240}, {-5/81, -51740/729}}}
     
106. {654, 1, {{5256, -6901128}, {-1695204, -259897428}}}
     
107. {658, 2, {{5288, -6985448}, {568835532278198658061318865859342568898404/1924789756820359242984200794995316775881, -32920580091258952117973526896643892652414947369879531172277257716/84445196033348822349274516428059623873224984950952653790971}, {-2401, 3164714}}}
     
108. {660, 1, {{5304, -7027800}, {-64921844/173889, -35812272632500/72511713}}}
     
109. {664, 1, {{5336, -7112888}, {-359610336/361, 6016559760768/6859}}}
     
110. {676, 2, {{5432, -7371224}, {466489/4840000, 1473500948237/10648000000}, {-20216/81, 246510712/729}}}
     
111. {686, 1, {{5512, -7590024}, {-10388/758641, 4791451644/660776311}}}
     
112. {690, 1, {{5544, -7678440}, {-235672500/94249, -99996090648300/28934443}}}
     
113. {692, 1, {{5560, -7722840}, {-2523472336029804618306645845/53860349231859847373133360169, -704982154609964674015328048386490486582207280/12499819668114993336962728370432736699477803}}}
     
114. {706, 2, {{5672, -8037224}, {2401/336400, -3169761049/195112000}, {-36/1681, -1433172/68921}}}
     
115. {714, 1, {{5736, -8219688}, {-29371476457739359572/122539665910468921, -14711837349803586866305424495700/42895826288674275429532781}}}
     
116. {718, 1, {{5768, -8311688}, {-4566626291129353781202675872288/5222186751318770502399574609, -474773771047549747263202699147152696889556405824/377379754781071388942264926979992904269273}}}
     
117. {724, 1, {{5816, -8450648}, {-614784437753409/938552126521, 864066176290791647757738/909258976100153069}}}
     
118. {728, 1, {{5848, -8543928}, {-345460757993344539935526087334135584536/26664583481268574325187941801482900605625, 32067444693327886551199931408398774418312488728387166660770312/4354138167453758145369452634187064449472432177813876117921875}}}
     
119. {732, 1, {{5880, -8637720}, {-4327904/352425529, 39538525918528/6616084455917}}}
     
120. {760, 2, {{6104, -9308600}, {-2292196, 378878500}, {961/784, -41156375/21952}}}
     
121. {762, 1, {{6120, -9357480}, {-17/49, 178942/343}}}
     
122. {774, 1, {{6216, -9653448}, {-3077659956/2468201761, -236164218582601092/122622731688241}}}
     
123. {776, 1, {{6232, -9703224}, {-154647628568/161170934521, -96021871857943438584/64703844543735181}}}
     
124. {782, 2, {{6280, -9853320}, {-3887703596027551016/198438187802481, 85474012581611695930361156216/2795360849879679573129}, {-1447220/978121, -2235074672820/967361669}}}
     
125. {812, 1, {{6520, -10621080}, {-184196557824006428603744/139672180358571723027169, -111582114797041308229316708563270752768/52199322844802431304415416023903697}}}
     
126. {816, 2, {{6552, -10725624}, {504/25, -4121208/125}, {-1053, 1721304}}}
     
127. {826, 2, {{6632, -10989224}, {-196/25, -1620892/125}, {16798590975828165504200/3871797428565361, -2780838896758834746422350190852360/240917828809845024265609}}}
     
128. {852, 1, {{6840, -11689560}, {-24441752/284698129, -664363276031192/4803711530617}}}
     
129. {874, 2, {{7016, -12299048}, {-3346911987660004/42521680348225, -38212369484310951003345844/277278137123928214625}, {-2656900/811801, 4185784896220/731432701}}}
     
130. {876, 1, {{7032, -12355224}, {-248378136/45954841, -2952484804009608/311527867139}}}
     
131. {882, 1, {{7080, -12524520}, {-89675787494593482561098389600149418396228704842820500/2927925040966503792399475931668211647942705510761, -269805139923722756810076384219303636661398131670094569280850811317972020750724300/5010025437919027236996162342940447749253178648692723901339539072411699509}}}
     
132. {886, 1, {{7112, -12638024}, {-513203716/900601, 864392151861692/854670349}}}
     
133. {892, 1, {{7160, -12809240}, {-2405378547170150561389840164023871553499000/35020634507790308582874090136642399412521, -804333482240516226034116690369946835691940198960179902135503941800/6553691824622282193476333143977254724534389793835287046634069}}}
     
134. {896, 1, {{7192, -12924024}, {-187809824330039849749891151197694617719473395412853929528/1624697429932303742838810195201608438364763806761289796321, 13053074223012108009173902576681697633466974369866581036714216735323701104782156325642504/65487549572547890747800950836465877690974316906261665852128985325087121600367274684881}}}
     
135. {912, 1, {{7320, -13388280}, {-216/25, 1972152/125}}}
     
136. {928, 1, {{7448, -13860728}, {-324/169, 7812468/2197}}}
     
137. {934, 2, {{7496, -14040008}, {-30817841071558085216871967862347661517736769753625250094244/3536845310159741358856517037093297676089365310046800278038129, -14135046986618262361769065329696440231513013866872957216688290126711470789272012100519438516/6651568967240720994318542148267022696433837346715345662355496576886379556441555684100260617}, {-3467044, 635377708}}}
     
138. {938, 1, {{7528, -14160168}, {-30209446196/29473679041, 9704461926569204124/5060011744079839}}}
     
139. {940, 2, {{7544, -14220440}, {-4/289, 79460/4913}, {7544/151321, -5978959480/58863869}}}
     
140. {942, 1, {{7560, -14280840}, {-420/169, 10285380/2197}}}
     
141. {952, 1, {{7640, -14584760}, {-11062332937832876728/85328276053021681, 6161995400514899282759228362696/24925246216307481410798921}}}
     
142. {954, 1, {{7656, -14645928}, {-2078289869739244356733015909224739450513773600/1078490706412589276789126280110459008297138729, -130144906455576369086797368607089875642451434126300033465864766280842240/35418047053081881188103007107239634454594152385680881244336701391317}}}
     
143. {966, 1, {{7752, -15015624}, {-1253915373550125325929567741477461834524379310308/18738863245838382266320487416224828698976798127681, -9831756064870030051531278328009054875975348638248830319745905530565116291476/81117557006785855185270779250071876073223958095129525337215754963302463329}}}
     
144. {970, 2, {{7784, -15139880}, {-13899042772164/1607454502583329, 232643875486225769194860/64447790716252098180017}, {-87461024/8121073689, -7411111636967360/731846797631613}}}
     
145. {976, 1, {{7832, -15327224}, {-342747515964623354876032241/7025209262693678426071325625, -51228313074680840201195437849088313342149902/588828602635833660571791291095848615921875}}}
     
146. {982, 2, {{7880, -15515720}, {-1482062760/145079477449, -499701131573380440/55259757403981957}, {-3943840/8404201, 22292990276480/24363778699}}}
     
147. {994, 2, {{7976, -15896168}, {-343069796945137870084/155506078110846479641, 8502233421663893656588884711291748/1939193080369699053180219544061}, {-1291518785000/93893987591881, -16049232143578261831400/909822505320679374971}}}
     
148. {996, 1, {{7992, -15960024}, {-888/109561, -55626984/36264691}}}
     
149. {1000, 1, {{8024, -16088120}, {-45395230454502763351961636504/2436593766529133091649625847769, -107213019898213331168666461156992414925539072520/3803423567716937825688136072659577147707805853}}}

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wayne

0<n<=10000以内,秩为3,且可以被PARI/Gp计算出全部生成元的情况是

1. 424,680,1336,1766,2390,2460,2866,3138,3310,3578,3678,3932,3984,4002,4154,4438,4440,4534,4696,4974,5120,5956,6266,6454,6524,6920,6976,7010,7286,7326,7626,7776,7904,8014,8194,8354,8376,8538,8874,9038,9390,9720

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然后有四个SageMath的计算不完整,不能确定秩是否为3

1. 5738, 6444, 7836, 9244

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然后扔给magma

1. SetClassGroupBounds("GRH");
   
2. for k in [5738, 6444, 7836, 9244] do
   
3.     E := EllipticCurve([0, 4*k^2 + 12*k - 3, 0, 32*(k+3), 0]);
   
4.     gens := Generators(E);
   
5.     print k, ",", #gens,",", gens;
   
6. end for;

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magma第一个超时了,2分钟不能返回结果.另外三个很爽快

1. 6444 , 4 , [ (51576 : -664969368 : 1), (-901616425303583282912048631286101945229976/74146335242380264442742177094931274142642225 :94834982674242693801623461044034157205313624972256490647210125233832/638461237980935221884766507226486803384138953071895831331861420375 : 1),(-11711127219256649119879256/623594852833184112065689 :119212603521748509300807454864335744618184/492440521750405560801423626991272387: 1), (-4357362296/922477648849 : 46320527272686537976/886000115180563993 : 1) ]
   
2. 7836 , 4 , [ (62712 : -983136024 : 1), (637/36 : -59911579/216 : 1),(-10021288011360700310923335237621081/10058363341224086229924493780081 :498188432801828184294506128337014404484688129860271118/31900021779883623997374779760342527601816000521 : 1), (-9481304/625 : 3715371088744/15625 : 1) ]
   
3. 9244 , 4 , [ (73976 : -1368038168 : 1), (-188414088342221489054162251256/1017796689491107885845429055225 : 3506604183293318135232367168862439638361363794072/1026813455087988547709785330080180450229365875 : 1), (-56/529 : 23718856/12167 :1), (1618180608487204/15423257190825649 :-3731292054185210090831779124/1915420587794314015349143 : 1) ]
   

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