|
|
Good curves for ECM
The paper
ECM using Edwards curves
by Bernstein, Birkner, Lange, and Peters
explains how these curves were found
and why these curves are useful.
x^2+y^2=1+dx^2y^2 with torsion group Z/12Z
| d |
some non-torsion points |
some torsion points |
a b e f |
| -24167/25 |
(5/23,-1/7)
|
(5/13,-1/13)
|
3 2 23 7
|
| -895973/27 |
(81/5699,-901/2501)
(117/133,-1/337)
|
(9/41,-1/41)
|
65 52 1729 56953;
13515 10812 4626453899 20280609
|
| -13391879/121 |
(11/6739,-29/33)
(11/589,-17/107)
(11/349,-13/137)
(13/137,-11/349)
(17/107,-11/589)
(29/33,-11/6739)
(451/549,-7/3355)
(121/122,-3/7747)
|
(11/61,-1/61)
|
78 65 58981 1781;
102 85 170221 1819;
174 145 5667499 957;
198 165 12078 2812161;
858 715 2370511 648791;
1122 935 2421089 1872431;
1722 1435 1102941 39478285;
1914 1595 1273767 62342489
|
| -5883882125/243 |
(729/34529,-85/8833)
(27/82445,-29/55)
(29/55,-27/82445)
|
(27/365,-1/365)
|
406 377 69336245 1595;
3654 3393 3488265 208008735;
10710 9945 748416075 60815205
|
| 1375/1024 |
(56/65,-44/5)
(8/17,-20/19)
(128/115,-3/5)
|
(4/5,-8/5)
|
15 -5 425 380;
36 -12 1035 3840;
231 -77 55055 10780
|
| 81289/15625 |
(5/4,-25/89)
|
(5/13,-25/13)
|
3 -2 4 89
|
| 4913/18225 |
(285/293,-153/569)
(35/17,-81/17)
(45/19,-3)
|
(15/17,-9/17)
|
12 3 57 27;
84 21 3213 833;
1292 323 1608863 3491953
|
| -268279/35721 |
(11/61,-567/643)
(651/5771,-468/493)
(567/643,-11/61)
|
(21/29,-9/29)
|
495 198 233409 489159;
1155 462 6214131 544621;
4030 1612 120937076 6159049
|
| 50531/46656 |
(36/325,-1938/1937)
(352/377,-729/481)
(138/107,-252/277)
(252/277,-138/107)
(1938/1937,-36/325)
|
(12/13,-18/13)
|
805 -161 482356 1025731;
990 -198 1007721 931216;
1615 -323 2502604 33906925;
2415 -483 6154386 6511806;
4845 -969 203441550 33785154
|
| -12907375/88209 |
(33/545,-153/190)
(891/3341,-115/403)
(153/190,-33/545)
|
(33/65,-9/65)
|
119 68 157505 3230;
3927 2244 12897390 46778985;
7245 4140 132554175 33785505
|
| 1140625/117649 |
(91/575,-539/475)
(29/25,-343/2025)
(35/109,-49)
(49,-35/109)
|
(7/25,-49/25)
|
20 -15 545 25;
140 -105 175 186935;
572 -429 904475 883025;
812 -609 248675 11921175
|
| 171875/186624 |
(1984/1985,-243/2165)
|
(24/25,-18/25)
|
3906 558 14953005 99867120
|
| 203125/196608 |
(396/445,-1856/1735)
(1856/1735,-396/445)
(2496/2545,-284/115)
|
(24/25,-32/25)
|
6699 -957 49400340 54793035;
12922 -1846 166781485 88312640;
13398 -1914 164379105 263468480
|
| -6967590689/295936 |
(748/15497,-56/419)
(17408/20217,-53/13771)
|
(68/293,-8/293)
|
1463 1155 16705766 709786;
16112 12720 113579106 2989518848
|
| 560947/352947 |
(63/58,-539/1189)
(21/19,-49/101)
(49/101,-21/19)
|
(21/29,-49/29)
|
5 -2 19 101;
105 -42 19089 19551;
165 -66 21054 117711
|
| -4885742359/385641 |
(138/1711,-4257/39007)
|
(69/269,-9/269)
|
12298 9460 765600638 73801244
|
| -148877/421875 |
(95/389,-2475/2579)
(2475/2579,-95/389)
|
(45/53,-25/53)
|
4389 1254 72439191 10241209;
7315 2090 51206045 111788875
|
| 41083561/1771561 |
(209/1891,-3509/2989)
(11/53,-121/4)
(121/4,-11/53)
|
(11/61,-121/61)
|
6 -5 53 4;
66 -55 44 70543;
3306 -2755 30216289 31291841
|
| -62258931486209/2876416 |
(424/18299,-316/34075)
|
(212/2813,-8/2813)
|
4345 4029 114204059 10767700
|
| -414078551/4515625 |
(1/10,-2125/2957)
(5/247,-425/433)
(425/5749,-35/43)
(425/433,-5/247)
|
(85/157,-25/157)
|
55 30 2165 30875;
187 102 1213511 7361;
385 210 1408505 37625;
935 510 6141250 251345
|
| 486256289/11390625 |
(75/653,-765/509)
(765/509,-75/653)
|
(15/113,-225/113)
|
408 -357 25959 5095359;
680 -595 943585 1081625
|
| 58512427/14480427 |
(117/373,-1183/967)
(1183/967,-117/373)
|
(39/89,-169/89)
|
168 -105 54831 60921;
2184 -1365 2375919 40154569
|
| -53081658551/15015625 |
(1705/83839,-425/667)
(3565/34323,-100/629)
(3875/33643,-61/425)
|
(155/493,-25/493)
|
828 598 3157716 332741;
3366 2431 266524181 1372019;
5490 3965 125185603 16203125
|
| 35975875/30118144 |
(520/569,-4802/577)
|
(56/65,-98/65)
|
5005 -1365 25371710 4875650
|
| -342703439/34000561 |
(119/841,-931/1031)
|
(119/169,-49/169)
|
228 95 303601 19589
|
| -921818375/42471289 |
(7847/17753,-980/2501)
(3059/3545,-147/1205)
|
(133/205,-49/205)
|
897 414 733815 1912335;
7670 3540 104742700 43529905
|
| 65743873/44289025 |
(115/79,-1331/1843)
|
(55/73,-121/73)
|
2024 -759 2418427 10724417
|
| 3207989881/47045881 |
(1159/191,-3971/33239)
|
(19/181,-361/181)
|
6710 -6039 1409771 1360505509
|
| -4610275679/62236321 |
(1817/4267,-2401/10081)
|
(161/289,-49/289)
|
8295 4424 115622899 62915521
|
| -525522625/126157824 |
(5772/11035,-504/865)
|
(156/205,-72/205)
|
4921 1813 40012910 16578590
|
| 138671875/130323843 |
(2223/1165,-3887/4055)
|
(117/125,-169/125)
|
4807 -874 11709415 33668665
|
| 14757432625/148035889 |
(575/5777,-8993/1007)
(299/7445,-529/485)
(529/485,-299/7445)
|
(23/265,-529/265)
|
156 -143 96785 81965;
1020 -935 1669553 427975;
3588 -3289 1885195 1177583095
|
| -636837799/191850201 |
(8/17,-41553/62033)
(3933/4537,-891/3341)
(41553/62033,-8/17)
|
(171/221,-81/221)
|
1512 540 2977584 2007666;
3542 1265 12626471 19441279;
9576 3420 170007174 56574096
|
| -1483680849359/259886641 |
(2303/44353,-203/823)
|
(329/1129,-49/1129)
|
5481 4060 261106111 8186381
|
| 5119323667/444107667 |
(207/131,-529/2279)
|
(69/269,-529/269)
|
39 -30 393 20511
|
| -50770053199/510714801 |
(2511/4931,-549/3301)
|
(279/521,-81/521)
|
3660 2013 18348251 16310241
|
| -3936492173867969/519110656 |
(4272/3007397,-80/323)
|
(712/7937,-32/7937)
|
1395 1275 225554775 58140
|
| 588384625/584043889 |
(39325/41059,-2873/2752)
|
(143/145,-169/145)
|
11220 -935 130526561 141521600
|
| -414563901477167/1061782225 |
(16625/53313,-49/10049)
|
(665/4537,-49/4537)
|
255 220 53313 251225
|
| -6798260327/1107225625 |
(275/313,-121/607)
|
(275/373,-121/373)
|
18 7 313 607
|
| -26597029199/1489882801 |
(253/259,-3509/69634)
(1073/1937,-1331/4069)
(3509/69634,-253/259)
|
(319/481,-121/481)
|
5060 2277 405200246 7928767;
8140 3663 95391439 61275071;
13340 6003 145285273 1068255194
|
| -12973390625/1514143744 |
(15200/157573,-2752/2873)
|
(304/425,-128/425)
|
5805 2365 291352477 12353900
|
| 1932101119/1936000000 |
(308/317,-25000/25421)
|
(220/221,-200/221)
|
3675 175 13868750 12456290
|
| 659873463625/2565726409 |
(925/533,-9583/187917)
|
(37/685,-1369/685)
|
665 -630 26117 32885475
|
| -482014149959/2977994041 |
(88/1037,-4961/7327)
(4961/7327,-88/1037)
|
(451/901,-121/901)
|
1144 660 1289552 2760494;
4264 2460 142942154 4806512
|
| -689280542159/8924769841 |
(13/113,-7267/10403)
(7267/10403,-13/113)
|
(559/1009,-169/1009)
|
364 195 135239 248261;
1204 645 8984291 447329
|
| 62925475961/18983603961 |
(2349/199,-11907/21733)
|
(189/389,-729/389)
|
10353 -6090 5938359 493491231
|
| 66043495387/23962599387 |
(217/2687,-8649/8599)
(8649/8599,-217/2687)
|
(279/521,-961/521)
|
420 -231 1523529 421351;
4340 -2387 13061881 560338919
|
| 35321542531/24716870656 |
(986/139,-50864/60989)
|
(272/353,-578/353)
|
15950 -5742 31216064 564209239
|
| -388958351375/40575253489 |
(697/755,-3179/25045)
|
(697/985,-289/985)
|
319 132 91355 275495
|
| -599811250919/44630365081 |
(301/1051,-4913/7435)
|
(731/1069,-289/1069)
|
3570 1547 36144941 6193355
|
| -26580408048254849/53557067776 |
(925696/1167967,-3/2747)
|
(1808/12833,-128/12833)
|
5808 5040 10511703 540082176
|
| 3924557083387/66493083387 |
(1431/54569,-457867/448633)
|
(159/1409,-2809/1409)
|
13692 -12225 1449843761 658144611
|
| 115613053375/142002356224 |
(26588/14915,-45056/38465)
(80/67,-23552/14473)
(23552/14473,-80/67)
|
(736/785,-512/785)
|
780 140 361825 343040;
4485 805 16303780 8321975;
14586 2618 115501760 122280235
|
| 122089230073/181343964025 |
(385/311,-12167/2971)
|
(805/877,-529/877)
|
7337 1518 41623307 8268293
|
| 695015111131/356257818075 |
(15375/31217,-149609/124766)
|
(615/953,-1681/953)
|
12460 -5785 247269857 277604350
|
| 1529005254361/1357269270361 |
(37/13,-31487/33683)
(74/73,-31487/75263)
(31487/33683,-37/13)
|
(851/949,-1369/949)
|
690 -161 158171 774709;
1110 -259 1246271 658489;
1380 -322 1776382 6924196
|
| -20034774365211173/1545938788827 |
(1357/22457,-14283/99467)
|
(4071/15929,-529/15929)
|
900 693 5457051 298401
|
| 1341892422625/1960495631329 |
(141/115,-29791/7145)
|
(1457/1585,-961/1585)
|
3627 744 10277895 1993455
|
| 3065462911001/2959263503001 |
(1131/1607,-56277/55277)
|
(1131/1181,-1521/1181)
|
1258 -185 2199983 2045249
|
| -444129411927679/12151422120321 |
(105633/306337,-231/569)
|
(3201/5249,-1089/5249)
|
15015 7392 495346929 143137071
|
| 13424713907683/15227477017600 |
(406/307,-67280/57229)
(67280/57229,-406/307)
|
(2320/2441,-1682/2441)
|
4830 770 17192000 14021105;
14007 2233 162644818 104823922
|
| -608428775842511/18752147640625 |
(13433/51592,-30625/56737)
|
(3535/5713,-1225/5713)
|
6460 3135 122531000 20482057
|
| 29735581036729/23279418765625 |
(192995/237617,-25/17)
|
(1595/1933,-3025/1933)
|
462 -143 237617 248897
|
| 1088769494428625/78285812197329 |
(2673/11185,-29791/13915)
|
(1023/4385,-8649/4385)
|
14508 -11439 333212335 314465085
|
| 89794825506079/89718784000000 |
(2000/1993,-4736/5015)
(4736/5015,-2000/1993)
|
(2960/2969,-3200/2969)
|
770 -30 626875 637760;
14245 -555 201902858 231943750
|
| 327721101310321/307568781694321 |
(497/177,-35287/36587)
|
(3479/3721,-5041/3721)
|
420 -77 60711 256109
|
| -13488069957887333/502169088896667 |
(31293/105533,-160003/307178)
|
(10431/16481,-3721/16481)
|
14964 7095 585391551 118877886
|
| 1403578139153875/1035243341090643 |
(99807/63265,-21218/26795)
|
(5253/6605,-10609/6605)
|
2926 -988 4808140 19345990
|
| -61620479060687234375/1910518904127289 |
(19787/492285,-50807/370555)
(158296/10635565,-2209/6305)
|
(19787/89725,-2209/89725)
|
936 748 21271130 100880;
5382 4301 260418765 8522765
|
| 2323107420282961/3559043436282961 |
(9559/74729,-6241/6259)
|
(9559/10441,-6241/10441)
|
100 21 74729 6259
|
| 59753578789817441/6785850023865441 |
(3519/15071,-23409/17359)
|
(3519/11969,-23409/11969)
|
88 -65 15071 17359
|
| 61842508572265625/10788090491627289 |
(51/5290,-2036583/2036135)
(2036583/2036135,-51/5290)
|
(4437/12125,-23409/12125)
|
7917 -5394 3483480870 177143745;
13923 -9486 103842885 56839626990
|
| 23889664618890409/20328521728515625 |
(10585/107,-390625/423463)
|
(9125/10477,-15625/10477)
|
14355 -3770 1939375 356132383
|
| 48555530519629489/51352262057640625 |
(85675/35083,-2645/2557)
|
(17135/17713,-13225/17713)
|
660 85 175415 319625
|
| 59121398597888561/68153847676800561 |
(514917/463733,-507/277)
|
(19071/20129,-13689/20129)
|
1260 207 1391199 605799
|
| 43131681866527446181375/19600932424529219682304 |
(3471616/5300791,-201640/63229)
|
(216976/359105,-645248/359105)
|
15180 -7540 265039550 161866240
|
| 2418954429246209275009/284081839323373126890625 |
(192061/196355,-2775125/13288277)
|
(960305/1108273,-555025/1108273)
|
5085 1360 24544375 66441385
|
x^2+y^2=1+dx^2y^2 with torsion group Z/2Z x Z/8Z
| d |
some non-torsion points |
some torsion points |
a b e f |
| 25921/83521 |
(289/299,7/23)
(17/19,17/33)
(13/7,289/49)
(323/161,561/161)
|
(17/7,17/7)
|
3 1 19 33;
357 119 249067 790993;
663 221 447083 140777;
1881 627 3331251 1917993
|
| 1681/707281 |
(319/403,551/901)
(122/123,841/6601)
|
(-29,-29)
(29,29)
|
627 418 1600313 2071399;
836 627 3200626 4142798;
7076 5307 182990667 1424614618;
10614 7076 365981334 2849229236
|
| 2307361/2825761 |
(943/979,1271/2329)
|
(41/31,41/31)
|
713 2852 21638837 38193271;
5704 713 43277674 76386542
|
| 23804641/62742241 |
(7921/8979,23/41)
(623/103,979/589)
(89/489,1513/1529)
(1691/4141,5607/5945)
|
(89/41,89/41)
|
51 85 141321 25993;
170 51 282642 51986;
231 385 87241 317471;
770 231 174482 634942;
1197 1995 34697439 15023015;
3990 1197 69394878 30046030;
6141 10235 422740299 664785767
|
| 418079809/442050625 |
(125/91,841/791)
(1015/437,1595/1541)
(2755/6223,9715/9779)
|
(145/127,145/127)
|
77 616 370139 830599;
145 1160 2219399 2867375;
1232 77 740278 1661198;
1273 10184 530765893 236524673;
2320 145 4438798 5734750
|
| 44182609/1766100625 |
(8405/15801,215/253)
(1025/158,697/25)
(6478/6647,25625/112999)
|
(-205/23,-205/23)
(205/23,205/23)
|
595 510 228310 53125;
1020 595 456620 106250;
12341 10578 1197858009 749791559;
23700 13825 1640978125 7052267590
|
| 779135569/1766100625 |
(1025/1032,41/265)
|
(205/103,205/103)
|
55 15 5160 33125
|
| 857376961/3373402561 |
(723/118,6989/3379)
|
(-241/89,-241/89)
(241/89,241/89)
|
696 957 595428 1763838;
957 348 297714 881919
|
| 1003559041/6975757441 |
(4913/377,51/19)
(4913/8611,731/869)
|
(-289/79,-289/79)
(289/79,289/79)
|
357 408 57681 280041;
816 357 115362 560082;
5117 5848 270669563 183584071;
11696 5117 541339126 367168142
|
| 3139697089/8653650625 |
(305/851,305/319)
(3721/7771,375/409)
|
(305/137,305/137)
|
13 4 851 319;
11895 3660 533284875 278505687
|
| 9200838241/20200652641 |
(18096/9793,62959/30191)
|
(377/193,377/193)
|
30060 8016 819350931 726033168
|
| 23226064801/43617904801 |
(8683/5767,4113/1663)
|
(457/257,457/257)
|
969 228 986157 600343
|
| 5149354081/236010384481 |
(28577/34343,527/943)
(41/89,11849/13319)
|
(-697/73,-697/73)
(697/73,697/73)
|
221 187 437257 226423;
374 221 874514 452846;
16523 13981 1353148543 2014767593
|
| 154542548161/1036488922561 |
(1009/15801,41369/41441)
|
(-1009/281,-1009/281)
(1009/281,1009/281)
|
533 615 26561481 1699081;
1230 533 53122962 3398162
|
| 294329035441/1998607065841 |
(89175/1813,22591/8659)
|
(-1189/329,-1189/329)
(1189/329,1189/329)
|
5700 6555 3926958 74034450;
6555 2850 1963479 37017225
|
| 1105511353489/5998703100625 |
(7825/12866,22223/27025)
|
(-1565/487,-1565/487)
(1565/487,1565/487)
|
7810 9585 648575060 479693750;
9585 3905 324287530 239846875
|
| 26597369569/6972900390625 |
(40625/2147,12155/389)
|
(1625/71,1625/71)
|
19635 14960 75078443 45464375
|
| 10166959513489/12625407900625 |
(9425/5704,15457/12865)
|
(1885/1409,1885/1409)
|
1435 5535 47942120 65933125;
11070 1435 95884240 131866250
|
| 7324606372801/80246641886401 |
(2993/3091,2993/11457)
|
(-2993/647,-2993/647)
(2993/647,2993/647)
|
34 35 6182 22914;
35 17 3091 11457
|
| 32009784282961/124596517122961 |
(86866/18259,8481/4001)
|
(3341/1241,3341/1241)
|
24882 9009 258492663 580153002
|
| 474352674970321/495066460507921 |
(89623/51987,33019/31961)
|
(4717/4207,4717/4207)
|
665 6118 48399897 80765447;
12236 665 96799794 161530894
|
| 656138522270401/1296864216024001 |
(6001/4139,6001/1441)
|
(6001/3271,6001/3271)
|
61 15 4139 1441
|
| 14841421194472525014961/40827573200405732248561 |
(449509/48927,449509/268159)
|
(449509/202129,449509/202129)
|
217 353 48927 268159;
706 217 97854 536318
|
| 1359201678908083521282341761/28355994025622165550609932161 |
(12976609/16551747,12976609/20594209)
|
(-12976609/2021231,-12976609/2021231)
(12976609/2021231,12976609/2021231)
|
1687 1560 16551747 20594209;
3120 1687 33103494 41188418
|
| 121421627844135414651728737921/764888338217537963142994811521 |
(29573281/31533721,29573281/79031041)
|
(-29573281/8509711,-29573281/8509711)
(29573281/8509711,29573281/8509711)
|
2201 2584 31533721 79031041;
5168 2201 63067442 158062082
|
Version
This is version 2009.12.11 of the goodcurves.html web page.
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