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Table 68 is a sequel to Table 20, Table 21, Table 66, Table 67 containing pKs equivalent to K001, K002, K020, K131 respectively. These are by far the most interesting examples of such groups of cubics. For completeness, the list of cubics in each of these groups is given again below.

The most remarkable other examples are highlighted in yellow. Most of the time, a symbolic substitution SS{a - > f(a)} maps the cubics in a same line to the cubics of another same line. Very often, f is a simple trigonometric function of the angles of ABC.

For example, SS{a -> a^2 SA} or SS{a -> sin2A} maps the group K002 to the group K044.

Table 20

{K001, K005, K060, K073, K095, K129a, K129b, K261a, K261b, K276, K278, K341a, K341b, K342a, K342b, K390, K419a, K419b, K420a, K420b, K636, K856, K859a, K859b, K867a, K867b, K1042, K1053a, K1053b, K1054a, K1054b}

Table 21

{K002, K004, K007, K033, K034, K099, K172, K175, K183, K184, K227A, K227B, K227C, K317, K318, K319, K343, K344, K345, K346, K362, K366, K410, K445, K576, K647, K750, K879, K880, K963, K964}


{K003, K037, K102, K374, K663}


{K006, K070a, K070b, K168, K170, K171, K233, K424a, K424b, K678, K707, K857, K1039, K1045, K1046, K1047}


{K008, K042, K043, K108, K273, K283, K284}

Table 66

{K020, K128, K252, K322, K354, K356, K421, K422, K423, K432, K532, K699, K738, K739, K743, K787, K788, K789, K861, K862, K863, K864, K865, K866, K985, K989, K990, K991, K992, K995, K998, K999, K1000, K1001, K1008, K1012, K1013, K1014, K1015, K1016, K1022, K1023, K1024, K1028, K1029, K1030, K1031, K1032, K1033, K1034, K1035, K1036, K1037}


{K021, K326}


{K029, K585, K586}


{K035, K554}


{K044, K045, K176, K350, K612, K621, K627, K646, K674, K919}


{K046a, K264a}


{K046b, K264b}


{K049, K146, K373}


{K050, K112, K416}


{K058, K134, K206}


{K059, K114, K472, K543, K860}


{K067, K439}


{K092, K485, K922}


{K097, K157}


{K101, K977}


{K109, K431, K610}


{K113, K475, K478}


{K130, K695}

Table 67

{K131, K132, K135, K136, K155, K251, K323, K673, K744, K766, K767, K768, K769, K770, K771, K772, K773, K774, K775, K868, K960, K961, K986, K987, K988, K993, K994, K996, K997, K1002, K1003, K1006, K1007, K1017, K1018, K1019, K1020, K1021, K1025, K1026, K1038, K1040, K1041}


{K133, K180}


{K140, K141, K161, K174, K177, K644, K836, K959, K968}


{K154, K179}


{K159, K167, K181, K411}


{K160, K368, K659}


{K169, K605, K1043}


{K199, K200, K202, K351, K363, K414, K632, K965}


{K201, K365, K747, K748, K761}


{K235, K236, K924}


{K237, K242}


{K239, K240}


{K243, K832}


{K253, K254, K320, K321, K379, K430, K901}


{K255, K256, K279, K489, K495, K528}


{K269, K334, K436}


{K294, K623, K980, K981, K982, K983, K984, K1011}


{K308, K760}


{K311, K312, K453, K454, K510, K717}


{K316, K672, K877, K1004, K1005}


{K329, K709}


{K335, K367}


{K336, K337}


{K340, K720}


{K355, K357, K380, K791}


{K364, K491, K497, K499, K500, K502}


{K377, K428}


{K434, K749}


{K455, K637}


{K490, K496, K611}


{K531, K533, K534, K535, K538, K539}


{K675, K677, K718, K776, K777, K778, K779, K780, K781, K782, K783, K784, K785, K786, K790, K1009, K1010}


{K949, K950}