D. J. Bernstein
Integer factorization

# Good curves for ECM

This is joint work with Peter Birkner, Tanja Lange, and Christiane Peters. See our paper ECM using Edwards curves for more information on how we found these curves 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