Elliptic Curves with n-isogenies by Twisting
  n j D a-Invariants Rank  Points
  11 -11*131^3 -97966 [0, 1, 0, -4609920947265, 3809810098100895007] 4 [ (378891 : 1455178912 : 1),
(1225037 : 30332668 : 1),
(1227706 : 25663737 : 1),
 (1230731 : 20633888 : 1) ]
  11 -11*131^3 1 [1, 1, 1, -30, -76] 0 [  ]
  11 -2^15 -64259 [ 0, 0, 0, -39244098145824, -97088780355971420016 ] 4 [ (23852400 : 111970343628 : 1),
 (322323144/25 : 4905512268228/125 : 1),
 (2863064688/289 : 108356670398196/4913 : 1),
 (352041355242828809761272/617215740005929 : -208864300713701556216173696528120676/15334001066384279050283 : 1) ]
  11 -2^15 1 [0, 0, 0, -9504, 365904] 1 [ (132 : 1188 : 1) ]
  11 -11^2 -1472782 [0, 1, 0, -87486501720801, 1231219400621544950911] 5 [ (-12939095 : 14033891784 : 1),
 (-10426870 : 31777736959 : 1),
 (-10138717 : 32802825248 : 1),
 (-3927015 : 38912979424 : 1),
 (14134290 : 53088439189 : 1) ]
  11 -11^2 1 [1, 1, 0, -2, -7] 0 [  ]
  14 -3^3*5^3 -883554 [0, 0, 0, -2213192847046860, 49277979898046961779088] 4 [ (20044773 : 113880988935 : 1),
 (13714124536168189/167832025 : -1399045913053132129972613/2174263883875 : 1),
(22908474873/64 : -3439124018032653/512 : 1),
(-18895086/25 : -28215212796288/125 : 1) ]
  14 -3^3*5^3 1 [0, 0, 0, -2835, -71442] 0 [  ]
  14 3^3*5^3*17^3 -4541835 [ 0, 0, 0, -12273817774498875, 523353327190225837089750 ] 5 [ (63580615 : 858913300 : 1),
(5521077681/100 : 118327734632079/1000 : 1),
(7689444315/121 : -3131289491400/1331 : 1),
(64433679 : -3915685908 : 1),
(3058471689/64 : -109742370689637/512 : 1) ]
  14 3^3*5^3*17^3 1 [0, 0, 0, -595, -5586] 0 [  ]
  15 -5^2/2 -1734994 [ 0, 0, 0, -2031887821524300, 415986955124092299795600 ] 5 [ (54495661889487904/1144942569 : -25318764599128856652903092/38741421707253 : 1),
(7770445186/1849 : 50755250114804384/79507 : 1),
 (1034332409495580/908209 : -33243825557146306002480/865523177 : 1),
(-75399555 : -374882539365 : 1),
(108036690 : -1207255073760 : 1) ]
  15 -5^2/2 1 [0, 0, 0, -675, -79650] 0 [  ]
  15 -5^2*241^3/2^3 -3163927 [1, 1, 0, -1256518025406400, 17143905458932364018600] 5 [ (1027154407497440749/43284386401 : -236580888737885615598312726/9005273306341649 : 1),
(-31181275 : 161282645385 : 1),
(26990665809738155/1237562041 : -464633073629593470898500/43536195040339 : 1),
(199640991790129/9884736 : 67425234225896150059/31077609984 : 1),
(8093230051159/80089 : -21765525609954880264/22665187 : 1) ]
  15 -5^2*241^3/2^3 1 [0, 0, 0, -162675, -25254450] 0 [  ]
  15 -5*29^3/2^5 1951555 [ 0, 0, 0, -14910539484067875, 844866803356652719721250 ] 4 [ (55086225 : -436646527200 : 1),
(-13179374995001/219961 :-127318277165210955818/103161709 : 1),
(16351825 : 778090263200 : 1),
(-658696919/25 : -138035145236896/125 : 1) ]
  15 -5*29^3/2^5 1 [0, 0, 0, -3915, 113670] 0 [  ]
  15 5*211^3/2^15 -39947 [ 0, 0, 0, 45455303614365, 53475829032149773470 ] 4 [ (-6436779951/5776 : 526118210838873/438976 : 1),
(-9420989/9 : -58828326724/27: 1),
(31360459 : 179781280172 : 1),
(-255549 : 6468622956 : 1) ]
  15 5*211^3/2^15 1 [0, 0, 0, 28485, -838890] 0 [  ]
  17 -17^2*101^3/2 703 [ 0, 0, 0, -1947437977635, 1046034687921458850 ] 5 [ (6650659794/8281 : 3539187807462/753571 : 1),
(612831 : 287646066 : 1),
(-189124575/169 : 2967092457480/2197 : 1),
 (804270 : -3313980 : 1),
(19666425 : -87000552360 : 1) ]
  17 -17^2*101^3/2 1 [1, 0, 1, -3041, 64278] 1 [ (1155 : 540 : 1) ]
  17 -17*373^3/2^17 11951 [ 0, 0, 0, -122264398180035, -583336689573060805950 ] 5 [ (78127905 : 683194521600 : 1),
(3913605921 : 244829588760576 : 1),
(34632456321/2401 : -3009034623154176/117649 : 1),
(17249511 : 49398166386 : 1),
(14415015 : 25486358130 : 1) ]
  17 -17*373^3/2^17 1 [1, 0, 1, -3041, 64278] 0 [  ]
  19 -2^15*3^3 182766 [0, 0, 0, -1269329608728, 550976951263356414] 5 [ (112465083/169 : 68777275293/2197 : 1),
(290441570253699/451095121 : -238993193058431214951/9580809274919 : 1),
(19471707026305/33918976 : -21168255653648131393/197544116224 : 1),
(118221060043/187489 : -2931220740023117/81182737 : 1),
(3461515/9 : -9340250869/27 : 1) ]
  19 -2^15*3^3 1 [0, 0, 1, -38, 90] 1 [ (0 : 9 : 1) ]
  21 -3^2*5^6/2^3 5107035 [ 0, 0, 0, -1956135486841875, 34898583036845276183250 ] 5 [ (46090864329/961 : 6754401321430608/29791 : 1),
(72966345 : 529760070000 : 1),
(3577364961/121 : 71773921582416/1331 :1),
(-94102671/4 : -2084572360017/8 : 1),
(793370191/361 : -1200034850307614/6859 : 1) ]
  21 -3^2*5^6/2^3 1 [0, 0, 0, -75, 262] 0 [  ]
  21 3^3*5^3/2 -2606345 [ 0, 0, 0, 305686541656125, 318689835765093245250 ] 5 [ (1313281045321/31329 : 1633662362419357256/5545233 : 1),
 (-169434243831/165649 : 149966639860061928/67419143 : 1),
(6804834 : -52095480198 : 1),
(307196844385/17161 : 241362919272914200/2248091 : 1),
(53992766961/169 : -12564778847008584/2197 : 1) ]
  21 3^3*5^3/2 1 [0, 0, 0, 45, -18] 0 [  ]
  21 -3^2*5^3*101^3/2^21 -531894503 [ 0, 0, 0, -34717516913297231259435, 5057797999381010687484847126015398 ] 5 [ (-4937020186071382517/21687649 : 3446361738314618898087612928/100999381393 : 1),
(31482936918276794059/40360609 : 172502619651850534309837788416/256410948977 : 1),
(-73465416801394637/519841 :31674169544494340324603960/374805361 : 1),
(265576128502369159073979/1048889367409 :120100054785610966455163454372450304/1074223192300029577 : 1),
(-86302937482923861/368449: -4115142372909599612227584/223648543 : 1) ]
  21 -3^2*5^3*101^3/2^21 1 [0, 0, 0, -122715, -33611274] 0 [  ]
  21 -3^3*5^3*383^3/2^7 4094 [ 0, 0, 0, -288873008460, 59759755711230096 ] 4 [ (2789260/9 : -11342872/27 : 1),
(310432 : -221788 : 1),
(224365 : 79001389 : 1),
(15245458/49 : 280192256/343 : 1) ]
  21 -3^3*5^3*383^3/2^7 1 [0, 0, 0, -17235, 870894] 0 [  ]
  27 -2^15*3*5^3 1058402 [0, 0, 0, -537703100929920, 4799460915663286150784] 5 [ (89167036/25 : -6762740028216/125 : 1),
(-8484287 : -93545512161 : 1),
(502930847804/37249 : -6670873267531000/7189057 : 1),
(2860797631081/155236 : -2073227727383158965/61162984 : 1),
(28201649561/2116 : -67573521504835/97336 : 1) ]
  27 -2^15*3*5^3 1 [0, 0, 0, -480, 4048] 0 [  ]
  37 -7*11^3 21880474 [ 0, 0, 0, -4976659705920307020, 4652644222913759472442719600 ] 5 [ (5233513994401/576 : 11644402263811465841/13824 : 1),
(1753799496 :-36317208774204 : 1),
(-30459000 : -69312336871140 : 1),
(60952749 : 65950963502787 : 1),
(4802717121760597123487076/2533328645545489 :5747175914622409721664974698465901616/127507960986737687172487 : 1) ]
  37 -7*11^3 1 [1, 1, 1, -8, 6] 1 [ (0 : 2 : 1) ]
  37 -7*137^3*2083^3 -3791 [ 0, 0, 0, -3875691624825195, 92869249364372376368550 ] 4 [ (35776959 : 1722559068 : 1),
(116326031937306/3236401 : -3041587442451636/5822285399 :1),
(323480455/9 : 195932260/27 : 1),
(6074346615/169 : -2104748820/2197 : 1) ]
  37 -7*137^3*2083^3 1 [0, 0, 0, -269675595, -1704553285050] 1 [ (42855 : 8089740 : 1) ]
  43 -2^18*3^3*5^3 18618 [ 0, 0, 0, -4769627754240, 4009361968209080448 ] 5 [ (11465692/9 : -689229800/27 : 1),
(-2356164 : 1472114808 : 1),
(-2072804 : 2233837288 : 1),
(136616759721/108241 : 111677074786773/35611289 : 1),
(3975783913/2704 : -58838243904757/140608 : 1) ]
  43 -2^18*3^3*5^3 1 [0, 0, 0, -13760, 621264] 1 [ (68 : -4 : 1) ]
  67 -2^15*3^3*5^3*11^3 37630 [ 0, 0, 0, -166976712848000, 830485352084480176000 ] 5 [ (31381929/4 : -14695709717/8 : 1),
(7708044 : -1177617928 : 1),
(2314236059436/310249 : 1055217218652184/172808693 : 1),
(6663316245961/893025 : 4485238331047291/843908625 : 1),
(7456065 : -21046175 : 1) ]
  67 -2^15*3^3*5^3*11^3 1 [0, 0, 0, -117920, 15585808] 1 [ (201 : 67 : 1) ]
  163 -2^18*3^3*5^3*23^3*29^3 1 [0, 0, 0, -34790720, 78984748304] 1 [ (3400 : 548 : 1) ]
             
             
             
             
             
             
   Elliptic Curves with n-isogenies Infinite Families and Twisting
  n j h or D a-Invariants Rank Points
  13   h = 101/60 [1, 0, 1, -288446980524976098, 59625981683441978523832756] 6 [ (456523722364424265245/1457044540561 : 159256817409135194357287843857/1758770781064912441 : 1),
(134553629994445/408321 : 156002514349515756263/260917119 : 1),
(12784560696995/40804 : 744869207101857591/8242408 : 1),
(9107238605 : 867637877336217 : 1), (3673081344470/3481 : 6263462261105596608/205379 : 1),
(20841355613570/57121 : 23464410003412338363/13651919 : 1) ]
  13 -28672/3 D = 70557 [ 0, 0, 0, -15054349712976, -24430364312064295536 ] 4 [ (398314540/81 : 3296805465428/729 : 1),
 (9488700 : 26211497292 : 1),
(153782505/4 : -1896893891829/8 : 1),
(18518985 : -77768349723 : 1) ]
  13 -28672/3 D = 1 [0, -1, 1, -2, -1] 0 [  ]
  13 -189/2 D = -1049 [ 0, 0, 0, -3119636835, -294524913592350 ] 4 [ (320994 : 178264962 : 1),
(38906361 : -242678034936 : 1),
(735590545/8649 : -5983925617720/804357 : 1),
(24363025/121 : -113253270920/1331 : 1) ]
  13 -189/2 D = 1 [1, -1, 0, -2, 6] 1 [ (-1 : 3 : 1) ]
  16   h = 187/40 [1, 1, 1, -30111109596200490, 201112119279475857520645] 4 [ (-159724087 : 1657108622043 : 1),
(-157449087 : 1687944602043 : 1),
(-61804087 : 1906869902043 : 1),
(2251905 : 1394030887355 : 1) ]
  16 -1/15 D = -1625560485 [ 0, 0, 0, -71346066040622751075, -37344705320499433158522629382750 ] 6 [ (105272109892569105/1530169 : -31863228225141109503751200/1892819053 : 1),
(19671035980095/121 : 86746724814411311850/1331 : 1),
(1399171831753815/6889 : 52174574389342485048150/571787 : 1),
(2576560830345 : -4135783464872798400 : 1),
(838744465817745/196 : -24290896971121829921025/2744 : 1),
(1371067644895 : 1605376625339641150 : 1) ]
  16 -1/15 D = 1 [1, 1, 1, 0, 0] 0 [  ]
  16 147281603041/5265 D = 59243810 [ 0, 0, 0, -500455990947572750700, 4309069892207051394932787054000 ] 7 [ (2337701498790/22201 : 6824623867464570955200/3307949 : 1),
(221133362294287905/17023876 : 621050556465135706380975/70240512376 : 1),
(2548931528084280/73441 : -106710840024916921271100/19902511 : 1),
(64840618034790/121 :-521672615919828057600/1331 : 1),
(45293557021340350470/3261209449 : 36041538534200256677456188800/186237888004043 : 1),
(725149278848765814/105370225 : 1180362945014439300693064512/1081625359625 : 1),
(14819187389590/1521 :35491391083772403200/59319 : 1) ]
  16 147281603041/5265 D = 1 [1, 0, 0, -110, 435] 0 [  ]
  18 -15625/28 D = -10533149 [ 0, 0, 0, -74889378802935675, -15965736745482965188337238 ] 6 [ (5691033382/9 : 371697916925204/27 : 1),
(142541497503669/17161 : 1700865928600933800816/2248091 : 1),
(80442291229/225 : 5805211057831808/3375 : 1),
(36769969714269 : 222966649722404326464 : 1),
(11975888901/25 : -952679632791024/125 : 1),
(830286919 : -22231391593114 : 1) ]
  18 -15625/28 D = 1  [1, 0, 1, -1, 0] 0 [  ]
  25 -4096/11 D = -203145767 [ 0, 0, 0, -17827863544807900848, -68811445261845457823111297904 ] 6 [ (1947526964340/169 : 2462298720453429012/2197 : 1),
(91839661188 : 27801396582298812 :1),
(46386254850109648/3916441 : 9109910599972536950044228/7750636739 : 1),
(65668431777828/6241 : 469954949192192766852/493039 : 1),
(50919744700 : 11447670118036636 : 1),
(23801404474246286976/26306641 : 116117887887614058884687655252/134926761689 : 1) ]
  25 -4096/11 D = 1 [0, -1, 1, 0, 0] 0 [  ]
  25 -19465109/22 D = 4817182 [ 0, 0, 0, -842698378523118060, -298043759642759592284430000 ] 6 [ (1507099290 : 43070563931040 : 1),
(228654654666/121 : -92766418635704544/1331 : 1),
(13716309095694474/1692601 : -1595615027246305628794272/2202073901 : 1),
(8229056949022785/6270016 :-459524858482363739402655/15700120064 : 1),
(75109501744 : 20583050027536988 : 1),
(8406519230436090/703921 : -768421024104745536017760/590589719 : 1) ]
  25 -19465109/22 D = 1 [1, 1, 1, -28, -69] 0 [  ]
             
             
             
             
             
             
  The case when n=163 revisited
  n j D a-Invariants Rank  Points
  163 -2^18*3^3*5^3*23^3*29^3 376085 [ 0, 0, 0, -4920796904905352000, 4201470077302436248011394000 ] 5 [ (5977856376867993364/4667485761 : -313700641780608402236612/318877959705759 : 1),
(158700684576026665/123609924 : 268828435223921335013675/1374295135032 : 1),
(1303661044 : 1425714466428 : 1),
(1020973353415081/784996 : -859311818356244702229/695506456 : 1),
(5117912865/4 : 619921407025/8 : 1) ]
  163 -2^18*3^3*5^3*23^3*29^3 1 [0, 0, 0, -34790720, 78984748304] 1 [ (3400 : 548 : 1) ]
             
             
             
             
             
             
  The cases when n=10 and n=12
  n j D a-Invariants Rank  Points
  10 2^-2 * 3^-1 * 11^-5 * 179^3 * 2699^3 -802314609 [ 0, 0, 0, -378767776027024717227, 2837315581689616108036312429530 ] 5 [ (41844275453799/25 : -213094783483409718/125 : 1),
(434321095553361/256 : 216829597114584193113/4096 : 1),
(188895273929308629/113569 : -833156864297925313455936/38272753 : 1),
(3780645016707189/2209 : 8984747174284626788736/103823 : 1),
(17255573048638553814/10310521 : -42505763779270400631212124/33107082931 : 1) ]
  10 2^-2 * 3^-1 * 11^-5 * 179^3 * 2699^3 1 [1, 0, 0, -10065, -389499] 0 [ ]
  12 -1 * 2^-12 * 3^-1 * 5^-3 * 11^3 * 59^3 -148243395 [ 0, 0, 0, -385087273215411336075, 9518604276230451280322858967750 ] 5 [ (-15169050045 : 3445230438963000 : 1),
(-7026382480005/289 : 10433204840541456000/4913 : 1),
(9547231851 : -2590813290100176 : 1),
(6052524226960059/2209 : 470862133751586945152448/103823 : 1),
(302287410585/4 :162385344066759525/8 : 1) ]
  12 -1 * 2^-12 * 3^-1 * 5^-3 * 11^3 * 59^3 1 [1,0,1,-14,-64] 0 [ ]