Let pK(Ω = p : q : r, P = u : v : w) be a pivotal cubic with isopivot P*, the barycentric quotient Ω ÷ P. We suppose Ω ≠ P^2 in order to discard pKs decomposed into the cevian lines of P. This cubic is anharmonically equivalent to K020 = pK(X6, X384) if and only if : ∑ a^2 (b^2 - c^2) (a^4 - b^2 c^2) [(b^2 + c^2)^2 - b^2 c^2] q r u^2 = 0. (E) (E) can be construed as follows : • X2, X3114, P^2 ÷ Ω are collinear, • Ω, Ω x X3114, P^2 are collinear, • P, P*, P* x X3114 are collinear, Recall that X3114 = t X3094, X3407 = g X3094, X3314 = t X3407. X x Y denotes the barycentric product of X and Y. For a given pole Ω, (E) shows that P must lie on a diagonal conic and for a given pivot P, (E) shows that Ω must lie on a circum-conic. *** The following table shows a large selection of cubics equivalent to K020. W denotes a Weak cubic otherwise it is Strong. See the related Table 67 and also Table 68.
 cubic Ω P n X(i) on the cubic for i < 3249 note K020 X6 X384 0 1, 3, 4, 32, 39, 76, 83, 194, 384, 695, 2896, 3224 (1) K128 X6 X385 0 1, 2, 6, 32, 76, 98, 385, 511, 694, 1423, 2319, 3186, 3225, 3229 (2) K252 X1691 X2 0 2, 3, 6, 83, 98, 171, 238, 385, 419, 1429, 1691, 2329 (4) K322 X1916 X694 0 2, 76, 141, 257, 297, 335, 384, 385, 694, 698, 1916, 2998 (5) K354 X694 X1916 0 2, 4, 6, 39, 256, 291, 511, 694, 1432, 1916 (6) K356 X3978 X76 -2 2, 69, 76, 290, 308, 350, 385, 694, 695, 732, 1909 (4) K421 X3407 X14617 2, 6, 83, 384, 458, 1031, 3114 K422 X6 X5999 1, 3, 4, 98, 147, 182, 262, 511 K423 X6 X3329 1, 2, 6, 39, 83, 182, 262 K432 X32 X1580 W 1 1, 6, 31, 75, 560, 1403, 1580, 1755, 1910, 1967, 2053 (2) K532 X8789 X694 -2 6, 25, 32, 237, 384, 385, 694, 733, 904, 1911, 2076, 3051 (6) K699 X385 X5989 98, 147, 1281, 1916 K738 X76 X3978 -2 2, 6, 75, 76, 290, 325, 698, 1502, 1916 (2) K739 X385 X6 0 2, 6, 83, 194, 239, 287, 385, 732, 894, 1916, 3225 (3) K743 X76 X9230 -2 6, 69, 75, 141, 264, 308, 1031, 1502, 2998 (1) K787 X9468 X9468 2 2, 6, 39, 232, 292, 694, 733, 893, 1691, 1915, 3224, 3229 (5) K788 X14602 X32 2 6, 32, 172, 248, 251, 385, 694, 695, 699, 1613, 1691, 1914 (3) K789 X1501 X1691 2 2, 6, 31, 32, 237, 699, 1501, 1691, 1976 (2) K861 X14602 X1 W 1 1, 31, 48, 82, 172, 1428, 1580, 1910, 1914, 1927, 1933, 2330 (4) K862 X385 X75 W -1 1, 63, 75, 239, 894, 1281, 1447, 1580, 1821, 1966, 1967, 2236, 3112 (4) K863 X1916 X1934 W 1 1, 38, 75, 92, 257, 335, 1581, 1925, 1926, 1934, 1959 (6) K864 X9468 X1581 W -1 1, 19, 31, 292, 893, 1431, 1581, 1755, 1964, 1965, 1966, 1967 (6) K865 X14603 X561 W -3 75, 304, 561, 1581, 1920, 1921, 1926, 1966 (4) K866 X14604 X1967 W -3 31, 560, 1580, 1582, 1922, 1923, 1927, 1967, 1973 (6) K985 X2 X1966 W -1 1, 2, 31, 75, 561, 1581, 1821, 1959, 1966, 2227, 3212 (2) K989 X1691 X31 W 1 1, 31, 82, 171, 238, 293, 1580, 1581, 1740, 1966, 2236 (3) K990 X694 X1967 W 1 1, 38, 75, 240, 256, 291, 1580, 1581, 1582, 1967, 2227, 3223 (5) K991 X8789 X1927 W 3 1, 31, 904, 1911, 1927, 1932, 1933, 1964, 1967 (5) K992 X3978 X1 W -1 1, 75, 336, 350, 1909, 1926, 1934, 1966, 3112 (3) K995 X18896 X1581 W -1 75, 334, 561, 1581, 1930, 1934, 1965, 1966 (5) K998 X2 X1965 W -1 2, 31, 38, 63, 92, 561, 1965, 3112, 3223 (1) K999 X32 X1582 W 1 6, 19, 48, 75, 82, 560, 1582, 1740, 1964 (1) K1000 X2 X5207 2, 4, 69, 147, 1031, 2896 K1001 X32 X6660 3, 6, 25, 2076 K1008 X1501 X1915 2 2, 25, 31, 184, 251, 1501, 1613, 1915, 3051 (1) K1012 X3094 X2 0 2, 3, 6, 76, 262, 982, 984, 3061, 3094, 3117 (7) K1013 X18898 X3407 0 2, 4, 6, 32, 182, 983, 985, 2344, 3114 (8) K1014 X3114 X3114 2 2, 6, 76, 183, 264, 870, 3114, 3224 (8) K1015 X3117 X1 1 1, 31, 48, 75, 1469, 2186, 2275, 2276, 3056, 3116 (7) K1016 X18899 X6 2 2, 6, 32, 184, 194, 263, 869, 3094, 3117 (7) K1022 X1502 X1926 W -3 1, 75, 76, 561, 1926, 1928, 1934 (2) K1023 X18896 X18896 2 2, 76, 141, 264, 325, 334, 1916 (6) K1024 X18901 X1502 -4 76, 305, 1502, 1916 (4) K1028 X1502 X1925 W -3 1, 76, 304, 1925, 1928, 1930, 1969 (1) K1029 X9233 X1933 W 3 1, 31, 32, 560, 1917, 1927, 1933 (2) K1030 X9233 X1932 W 3 1, 32, 1917, 1923, 1932, 1973 (1) K1031 X3407 X3113 W 1 1, 31, 75, 92, 3113 (8) K1032 X3314 X75 W -1 1, 63, 75, 561, 3116 (7) K1033 X18902 X6 2 6, 32, 184, 251, 1691, 1976, 2210 (4) K1034 X18903 X9468 -4 32, 1501, 1691, 1915, 1974 (6) K1035 X18904 X2 W 2, 10, 37, 238, 1581, 1921, 2887, 3061 K1036 X18905 X2 W 2, 171, 226, 982, 1214, 1215, 1920, 2887 K1037 X2 X18906 2, 4, 69, 194, 263, 2998, 3212
 Notes : each number refers to cubics with equations of the same type and with the same color in the table. Furthermore, Kxxxx(n+1) = X(1) x Kxxxx(n) with exceptions K354(n) and K1023(n) for which Kxxxx(n-1) = X(1) x Kxxxx(n). Note that X(1916) is the isotomic conjugate of X(385). (1) : K020(n) = pK(X1^(2n+2), X1^n x X384) (2) : K128(n) = pK(X1^(2n+2), X1^n x X385) (3) : K739(n) = pK(X1^(2n) x X385, X1^(n+2)) (4) : K252(n) = pK(X1^(2n+2) x X385, X1^n) (5) : K322(n) = pK(X1^(2n) x X1916, X1^(n+2) x X1916) (6) : K354(n) = pK(X1^(2-2n) x X1916, X1^(-n) x X1916) (7) : K1012(n) = pK(X1^(2n) x X3094, X1^n) (8) : K1013(n) = pK(X1^(2-2n) x X3407, X1^(-n) x X3407) Remarks : • these equations clearly show that the cubics above are always strong for n even and weak for n odd. • when Ω = X2 (resp. P = X2), the complement (resp. anticomplement) of pK(Ω, P) is another pK equivalent to K020. *** Additional data by Peter Moses The following table shows other cubics passing through at least 10 ETC centers. Most of them are simple barycentric products, see column 3.
 Ω, P X(i) on the cubic pK(Ω, P) for i <18859 products / notes 394, 12215 3, 63, 69, 147, 184, 194, 287, 305, 8858, 12215 X(69) x K128 1916, 8842 2, 262, 325, 427, 1916, 3329, 4518, 5999, 7249, 8842 2052, 17984 4, 25, 92, 264, 297, 2998, 9473, 16081, 17984, 18022 X(264) x K128 2207, 419 4, 19, 25, 232, 264, 419, 1974, 3224, 6531, 17980 X(4) x K128 3124, 804 512, 523, 661, 669, 804, 850, 882, 2395, 3569, 18010 X(523) x K128 3314, 325 2, 147, 262, 305, 325, 1916, 3314, 3705, 7179, 9865 note 1 3407, 8840 2, 25, 98, 183, 385, 3407, 5989, 5999, 8840, 9473 note 1 9427, 5027 512, 523, 669, 798, 881, 2422, 2491, 5027, 9426, 9429, 17997 X(512) x K128 14604, 8789 6, 32, 1922, 2211, 3051, 7104, 8789, 9468, 14602, 14946 X(9468) x K739 17493, 7018 2, 8, 238, 256, 350, 4388, 7018, 7261, 17280, 17493 X(257) x K862
 note 1 : these two cubics are isotomic transforms from one another.