Euler Coordinates. If P = (u : v : w) (barycentric coordination), then the Euler coordinates (xe, ye), of P, are defined by

Euler Coordinates The Euler coordinate system is introduced here as an orthogonal coordinate system in which the Euler line is the x-axis, the orthic axis is the y-axis, and the origin is X(468). The positive direction on the x-axis is that of the vector X(468)-to-X(2) ; the positive direction of y-axis is that of the vector X(468)-to-X(16272) when (b-c)(c-a)(a-b) is positive, otherwise that of the vector X(468)-to-X(16309). If P = (u : v : w) (barycentric coordination), then the Euler coordinates (xe, ye), of P, are defined by xe = S A u + S B v + S C w ye = -S A (S B -S C )u - S B (S C -S A )v - S C (S A -S B )w. Here xe = kd, where k = (E-8F) 1/2 = 6 X(2)X(3) and d is the signed distance from P to the orthic axis. Likewise, ye = ksd, where d is the distance from the Euler line. If P is a triangle center, X, then there exist symmetric function G(a,b,c) and H(a,b,c) and nonsymmetric function K(a,b,c) such that X = G(a, b,c)s 2 +H(a,b,c)+K(a,b,c), Let N = (3G+H)S 2 +$K$, the Euler coordinates of X are then given by xe = [(EG+FG+3FH)S 2 +$K$]/N ye = - [$S A (S B -S C )K$]/N In particular, for X on the Euler line, K(a,b,c) = 0 and (G, H) are the Shinagawa coefficients for X. If X is not on the Euler line, then X is classed according to the denominator N shown just above. If X is indexed in ETC as X(n), then the Euler coordinates of X are written as ( xe(n), ye(n) ). As an example of a class, Table 1 Euler coordinates (xe(n), 0) of X(n) in N=S 2 -class, on the Euler line, Shinagawa coefficients (G(n), H(n)), and GXe(n) = xe(2)xe(n) /(E-8F) X(n) xe(n) (G(n), H(n))S 2 GXe(n) X(2) (E+F)/3 (1, 0)/3 0

X(3) (E-2F)/2 (1, -1)/2 1/6 X(4) 3F (0, 1) - 1/3 X(5) (E+4F)/4 (1, 1)/4-1/12 X(20) (E-5F) (1, -2) 2/3 X(30) (1, -3) ---- X(140) 3E/8 (3, -1)/8 1/24 X(376) (2E-7F)/3 (2, -3)/3 1/3 X(381) (E+10F)/6 (1, 3)/6-1/6 X(382) (-E+14F)/2 (-1, 5)/2-5/6 X(546) (E+16F)/8 (1, 5)/8-5/24 X(547) (7E+16F)/24 (7, 3)/24-1/24 X(548) (5E-16F)/8 (5, -7)/8 7/24 X(549) (5E-4F)/12 (5, -3)/12 1/12 X(550) 3(E-4F)/4 (3, -5)/4 5/12 X(631) (2E-F)/5 (2, -1)/5 1/15 X(632) (7E+4F)/20 (7, -1)/20 1/60 X(1656) 3(E+2F)/10 (3, 1)/10-1/30 X(1657) 3(E-6F)/2 (3, -7)/2 7/6 X(3090) (2E+5F)/7 (2, 1)/7-1/21 X(3091) (E+7F)/5 (1, 2)/5-2/15 X(3146) (-E+11F) (-1, 4) - 4/3 X(3522) (3E-9F)/5 (3, -4)/5 4/15 X(3523) 3(E-F)/7 (3, -2)/7 2/21 X(3524) (4E-5F)/9 (4, -3)/9 1/9 X(3525) (4E+3F)/11 (4, -1)/11 1/33 X(3526) (5E+2F)/14 (5, -1)/14 1/42 X(3528) (4E-11F)/7 (4, -5)/7 5/21 X(3529) (2E-13F) (2, -5) 5/3 X(3530) (7E-8F)/16 (7, -5)/16 5/48 X(3533) (6E+3F)/17 (6, -1)/17 1/51 X(3534) (5E-22F)/6 (5, -9)/6 1/2 X(3543) (-E+17F)/3 (-1, 6)/3-2/3 X(3544) (4E+19F)/17 (4, 5)/17-5/51 X(3545) (2E+11F)/9 (2, 3)/9-1/9 X(3627) (-E+20F)/4 (-1, 7)/4-7/12

X(3628) (5E+8F)/16 (5, 1)/16-1/48 X(3830) (-E+26F)/6 (-1, 9)/6-1/2 X(3832) (E+13F)/7 (1, 4)/7-4/21 X(3839) (E+19F)/9 (1, 6)/9-2/9 X(3843) (E+22F)/10 (1, 7)/10-7/30 X(3845) (E+28F)/12 (1, 9)/12-1/4 X(3850) 3(E+8F)/16 (3, 7)/16-5/48 X(3851) (3E+18F)/14 (3, 5)/14-5/42 X(3853) (-E+32F)/8 (-1, 11)/8-11/24 X(3854) 3(E+9F)/17 (3, 8)/17-8/51 X(3855) (2E+17F)/11 (2, 5)/11-5/33 X(3856) (5E+56F)/32 (5, 17)/32-17/96 X(3857) (5E+44F)/28 (5, 13)/28-13/84 X(3858) 3(E+12F)/20 (3, 11)/20-11/60 X(3859) (7E+64F)/40 (7, 19)/40-19/120 X(3860) (7E+88F)/48 (7, 27)/48-9/48 X(3861) (E+40F)/16 (1, 13)/16-13/48 X(5054) (7E-2F)/18 (7, -3)/18 1/18 X(5055) (5E+14F)/18 (5, 3)/18-1/18 X(5056) 3(E+3F)/11 (3, 2)/11-2/33 X(5059) 3(E-7F) (3, -8) 8/3 X(5066) (5E+32F)/24 (5, 9)/24-1/8 X(5067) (4E+7F)/13 (4, 1)/13-1/39 X(5068) 3(E+5F)/13 (3, 4)/13-4/39 X(5070) (7E+10F)/22 (7, 1)/22-1/66 X(5071) (4E+13F)/15 (4, 3)/15-1/15 X(5072) (5E+26F)/22 (5, 7)/22-7/66 X(5073) 3(-E+10F)/2 (-3, 11)/2-11/6 X(5076) (-E+38F)/10 (-1, 13)/10-13/30 X(5079) (7E+22F)/26 (7, 5)/26-5/78 X(7486) (5E+11)/17 (5, 2)/17-2/51 X(8703) (7E-20F)/12 (7, -9)/12 1/4 X(10109) (13E+40F)/48 (13, 9)/48-1/16 X(10124) (17E+8F)/48 (17, -3)/48 1/48 X(10299) (6E-9F)/13 (6, -5)/13 5/39

X(10303) (5E-F)/13 (5, -2)/13 2/39 X(10304) (5E-13F)/9 (5-6)/9 2/9 X(11001) (4E-23F)/3 (4, -9)/3 1 X(11539) (13E+4F)/36 (13, -3)/36 1/36 X(11540) (35E+8F)/96 (35, -9)/96 1/32 X(11541) (-4E+35F) (-4, 13) - 13/3 X(11737) (11E+56F)/48 (11, 15)/48-5/48 X(11812) (19E-8F)/48 (19, -9)/48 1/16 X(12100) (11E-16F)/24 (11, -9)/24 1/8 X(12101) (-E+80F)/24 (-1, 27)/24-3/8 X(12102) (-E+56F)/16 (-1, 19)/16-19/48 X(12103) (7E-32F)/8 (7, -13)/8 13/24 X(12108) (13E-8F)/32 (13, -7)/32 7/96 X(12811) (7E+40F)/32 (7, 11)/32-11/96 X(12812) (11E+32F)/40 (11, 7)/40-7/120 X(14093) (17E-46F)/30 (17, -21)/30 7/30 X(14269) (E+46F)/18 (1, 15)/18-5/18 X(14869) (11E-4F)/28 (11, -5)/28 5/84 X(14890) (55E-8F)/144 (55, -21)/144 7/144 X(14891) (23E-40F)/48 (23, -21)/48 7/48 X(14892) (17E+80F)/72 (17, 21)/72-7/72 X(14893) (E+64F)/24 (1, 21)/24-7/24 X(15022) (5E+17F)/19 (5, 4)/19 4/57 X(15640) (-5E+49F)/3 (-5, 18)/3-2 X(15681) (7E-38F)/6 (7, -15)/6 5/6 X(15682) (-2E+25F)/3 (-2, 9)/3-1 X(15683) (5E-31F)/3 (5, -12)/3 4/3 X(15684) (-5E+58F)/6 (-5, 21)/6-7/6 X(15685) (11E-70F)/6 (11, -27)/6 3/2 X(15686) (11E-52F)/12 (11, -21)/12 7/12 X(15687) (-E+44F)/12 (-1, 15)/12-5/12 X(15688) (11E-34F)/18 (11, -15)/18 5/18 X(15689) (13E-50F)/18 (13, -21)/18 7/18 X(15690) (17E-64F)/24 (17, -27)/24 3/8 X(15691) (19E-80F)/24 (19, -33)/24 11/24

X(15692) (7E-11F)/15 (7, -6)/15 2/15 X(15693) (13E-14F)/30 (13, -9)/30 1/10 X(15694) (11E+2F)/30 (11, -3)/30 1/30 X(15695) (19E-62F)/30 (19, -27)/30 3/10 X(15696) (7E-26F)/10 (7, -11)/10 11/30 X(15697) (11E-43F)/15 (11, -18)/15 2/5 X(15698) (10E-17F)/21 (10, -9)/21 1/7 X(15699) (11E+20F)/36 (11, 3)/36-1/36 X(15700) (19E-26F)/42 (19, -15)/42 5/36 X(15701) (17E-10F)/42 (17, -9)/42 1/14 X(15702) (8E-F)/21 (8, -3)/21 1/21 X(15703) (13E+22F)/42 (13, 3)/42-1/42 X(15704) (5E-28F)/4 (5, -11)/4 11/12 X(15705) (13E-23F)/27 (13, -12)/27 4/27 X(15706) (25E-38F)/54 (25, -21)/54 7/54 X(15707) (23E-22F)/54 (23, -15)/54 5/54 X(15708) (11E-7F)/27 (11, -6)/27 2/27 X(15709) (10E+F)/27 (10, -3)/27 1/27 X(15710) (14E-31F)/27 (14, -15)/27 5/27 X(15711) (29E-52F)/60 (29, -27)/60 3/20 X(15712) (9E-12F)/20 (9, -7)/20 7/60 X(15713) (23E-4F)/60 (23, -9)/60 1/20 X(15714) (31E-68F)/60 (31, -33)/60 11/60 X(15715) (16E-29F)/33 (16, -15)/33 5/33 X(15716) (31E-50F)/66 (31, -27)/66 3/22 X(15717) (5E-7F)/11 (5, -4)/11 4/33 X(15718) (29E-34F)/66 (29, -21)/66 7/66 X(15719) (14E-13F)/33 (14, -9)/33 1/11 X(15720) 3(3E-2F)/22 (9, -5)/22 5/66 X(15721) (13E-5F)/33 (13, -6)/33 2/33 X(15722) (43E-38F)/102 (43, -27)/102 1/34 X(15723) (23E+14F)/66 (23, -3)/66 1/66 X(15759) (25E-56F)/48 (25, -27)/48 3/16 X(16239) (11E+8F)/32 (11, -1)/32 1/96

Table 2 Euler coordinates (xe(n), ye(n)) of X(n) in N={(E+F)+(3) 1/2 S}- class X(n) xe(n)*n ye(n)*n/(s B -S C )(S C -S A )(S A -S B ) X(13) [{(3) 1/2 (E+4F)S}+6S 2 ]/6-1/2 X(15) [{(3) 1/2 (E-2F)S}+6S 2 ]/2-1/2 X(298) [(E+F){3(E+F)+(3) 1/2 S}-6S 2 ]/3 1 X(396) [(E+F){3(E+F)+2(3) 1/2 S}+6S 2 ]/6-1/2 X(616) [{3(E+F) 2 +(3) 1/2 (2E-7F)S}-6S 2 ]/3 1 X(618) [{6(E+F) 2 +(3) 1/2 (5E-4F)S}- 6S 2 ]/12-1/2 X(621) {(E+F) 2 +2(3) 1/2 FS}-2S 2 1 X(623) [{(E+F) 2 +(3) 1/2 (E+4F)S} -2S 2 ]/4 1/4 X(3180) [-(E+F){3(E+F)-(3) 1/2 (2E-7F)S}+12S 2 ]/3-2 X(5318) 3(3) 1/2 FS+S 2-1/2 X(5459) [{2(E+F) 2 +3(3) 1/2 (E+4F)S}+6S 2 ]/12-1/4 X(5463) [{4(E+F) 2 +3(3) 1/2 (E-2F)S}-6S 2 ]/6-1/2 X(5473) [6(E+F)(E-2F)+(3) 1/2 (5E-22F)S}-6S 2 ]/6 1/2 X(5478) [{18(E+F)F+{(3) 1/2 (E+28F)S}+6S 2 ]/12-1/4 X(5611) [(E-2F){-(E+F)+(3) 1/2 S}+4S 2 ]/2-1 X(5617) [(E+F){3(E+2F)+2(3) 1/2 S}-6S 2 ]/6 1/2 X(6669) [{6(E+F) 2 +(3) 1/2 (7E+16F)S}+6S 2 ]/24-1/8 X(6671) [{2(E+F) 2 +(3) 1/2 ES}+2S 2 ]/8-1/8 X(6770) [(E+F){-9F+(3) 1/2 S}+6S 2 ]/3-1 X(6771) [(E+F){3(2E-F)+4(3) 1/2 S}+6S 2 ]/12-1/4 X(7684) [{6(E+F)F+(3) 1/2 (E+4F)S}+2S 2 ]/4-1/4 X(11542) [{(3) 1/2 (E+4F)S}+4S 2 ]/4-1/2 X(13103) [{9(E+F)F- (3) 1/2 (E-26F)S}+6S 2 ]/6-1 X(13350) [{(E+F)(2E-F)+(3) 1/2 (E-5F)S}+2S 2 ]/4 1/4 X(14539) [{(E-2F){2(E+F)-(3) 1/2 S}-2S 2 ]/2 1/2 X(16001) [{-(E+F)(E-2F)+12(3) 1/2 FS}+6S 2 ]/4-3/4 Table 3 Euler coordinates (xe(n), ye(n)) of X(n) in N={(E+F)-(3) 1/2 S}- class X(n) xe(n)*n ye(n)*n/(s B -S C )(S C -S A )(S A -S B ) X(14) [-{(3) 1/2 (E+4F)S}+6S 2 ]/6-1/2

X(16) [{-(3) 1/2 (E-2F)S}+6S 2 ]/2-1/2 X(299) [(E+F){3(E+F)-(3) 1/2 S}-6S 2 ]/3 1 X(395) [(E+F){3(E+F)-2(3) 1/2 S}+6S 2 ]/6-1/2 X(617) [{3(E+F) 2 -(3) 1/2 (2E-7F)S}-6S 2 ]/3 1 X(619) [{6(E+F) 2 -(3) 1/2 (5E-4F)S}- 6S 2 ]/12-1/2 X(622) {(E+F) 2-2(3) 1/2 FS}-2S 2 1 X(624) [{(E+F) 2 -(3) 1/2 (E+4F)S} -2S 2 ]/4 1/4 X(3181) [-(E+F){3(E+F)+(3) 1/2 (2E-7F)S}+12S 2 ]/3-2 X(5321) -3(3) 1/2 FS+S 2-1/2 X(5460) [{2(E+F) 2-3(3) 1/2 (E+4F)S}+6S 2 ]/12-1/4 X(5464) [{4(E+F) 2-3(3) 1/2 (E-2F)S}-6S 2 ]/6-1/2 X(5474) [6(E+F)(E-2F)-(3) 1/2 (5E-22F)S}-6S 2 ]/6 1/2 X(5479) [{18(E+F)F-(3) 1/2 (E+28F)S}+6S 2 ]/12-1/4 X(5615) [(E-2F){-(E+F)- (3) 1/2 S}+4S 2 ]/2-1 X(5613) [(E+F){3(E+2F)-2(3) 1/2 S}-6S 2 ]/6 1/2 X(6670) [{6(E+F) 2 - (3) 1/2 (7E+16F)S}+6S 2 ]/24-1/8 X(6674) [{2(E+F) 2 - (3) 1/2 ES}+2S 2 ]/8-1/8 X(6773) [(E+F){-9F- (3) 1/2 S}+6S 2 ]/3-1 X(6774) [(E+F){3(2E-F)- 4(3) 1/2 S}+6S 2 ]/12-1/4 X(7685) [{6(E+F)F- (3) 1/2 (E+4F)S}+2S 2 ]/4-1/4 X(11543) [{-(3) 1/2 (E+4F)S}+4S 2 ]/4-1/2 X(13102) [{9(E+F)F+(3) 1/2 (E-26F)S}+6S 2 ]/6-1 X(13349) [{(E+F)(2E-F)- (3) 1/2 (E-5F)S}+2S 2 ]/4 1/4 X(14538) [{(E-2F){2(E+F)+(3) 1/2 S}-2S 2 ]/2 1/2 X(16002) [{-(E+F)(E-2F)-12(3) 1/2 FS}+6S 2 ]/4-3/4 Table 4 Euler coordinates (xe(n), ye(n)) of X(n) in N=$a$-class X(n) xe(n)*$a$ ye(n)*$a$/{-$as A (S B -S C )$} X(1) $as A $ 1 X(8) (E+F)$a$-2$aS A $ - 2 X(10) {(E+F)$a$-$aS A $}/2-1/2 X(40) (E-2F)$a$-$aS A $ - 1 X(145) -(E+F)$a$+4$aS A $ 4

X(165) {2(E-2F)$a$-$aS A $}/3-1/3 X(355) {(E+4F)$a$-2$aS A $}/2-1 X(551) {(E+F)$a$+3$aS A $}/6 1/2 X(944) -3F$a$+2$aS A $ 2 X(946) {3F$a$+$aS A $}/2 1/2 X(962) (-E+5F)$a$+2$aS A $ 2 X(1125) {(E+F)$a$+$aS A $}/4 1/4 X(1385) {(E-2F)$a$+2$aS A $}/4 1/2 X(1482) {(-E+2F)$a$+4$aS A $}/2 2 X(1483) {-(E+4F)$a$+8$aS A $}/4 2 X(1698) {2(E+F)$a$-$aS A $}/5-1/5 X(1699) {6F$a$+$aS A $}/3 1/3 X(3241) {-(E+F)$a$+6$aS A $}/3 2 X(3244) {-(E+F)$a$+5$aS A $}/2 5/2 X(3576) {(E-2F)$a$+$aS A $}/3 1/3 X(3579) {3(E-2F)$a$-2$aS A $}/4-1/2 X(3616) {(E+F)$a$+2$aS A $}/5 2/5 X(3617) {3(E+F)$a$-4$aS A $}/5-4/5 X(3621) 3(E+F)$a$-8$aS A $ - 8 X(3622) {(E+F)$a$+4$aS A $}/7 4/7 X(3623) {-(E+F)$a$+8$aS A $}/5 8/5 X(3624) {2(E+F)$a$+$aS A $}/7 1/7 X(3625) {3(E+F)$a$-7$aS A $}/2-7/2 X(3626) {3(E+F)$a$-5$aS A $}/4-5/4 X(3632) 2(E+F)$a$-5$aS A $ - 5 X(3633) -2(E+F)$a$+7$aS A $ 7 X(3634) {3(E+F)$a$-$aS A $}/8-1/8 X(3635) {-(E+F)$a$+7$aS A $}/4 7/4 X(3636) {(E+F)$a$+5$aS A $}/8 5/8 X(3653) {(5E-4F)$a$+6$aS A $}/18 1/3 X(3654) {(5E-4F)$a$-6$aS A $}/6-1 X(3655) {(E-8F)$a$+6$aS A $}/6 1 X(3656) {-(E-8F)$a$+6$aS A $}/6 1 X(3679) {2(E+F)$a$-3$aS A $}/3-1 X(3817) {(E+7F)$a$+$aS A $}/6 1/6

X(3828) {5(E+F)$a$-3$aS A $}/12-1/4 X(4297) {(E-5F)$a$+$aS A $}/2 1/2 X(4301) {-(E-5F)$a$+3$aS A $}/2 3/2 X(4668) {4(E+F)$a$-7$aS A $}/5-7/5 X(4669) {5(E+F)$a$-9$aS A $}/6-3/2 X(4677) {4(E+F)$a$-9$aS A $}/3-3 X(4678) {5(E+F)$a$-8$aS A $}/7-8/7 X(4691) {5(E+F)$a$-7$aS A $}/8-7/8 X(4701) {5(E+F)$a$-11$aS A $}/4-11/4 X(4745) {7(E+F)$a$-9$aS A $}/12-3/4 X(4746) {7(E+F)$a$-13$aS A $}/8-13/8 X(4816) {6(E+F)$a$-13$aS A $}/5-13/5 X(5493) {3(E-3F)$a$-3$aS A $}/2-3/2 X(5550) {3(E+F)$a$+2$aS A $}/11 2/11 X(5587) {(E+4F)$a$-$aS A $}/3-1/3 X(5603) {3F$a$+2$aS A $}/3 2/3 X(5657) {(2E-F)$a$-2$aS A $}/3-2/3 X(5690) {3E$a$-4$aS A $}/4-1 X(5691) 6F$a$-$aS A $ - 1 X(5731) {(E-5F)$a$+2$aS A $}/3 2/3 X(5734) {-(E-5F)$a$+6$aS A $}/5 6/5 X(5790) {3(E+2F)$a$-4$aS A $}/6-2/3 X(5818) {(2E+5F)$a$-2$aS A $}/5-2/5 X(5881) (E+4F)$a$-3$aS A $ - 3 X(5882) {-3F$a$+3$aS A $}/2 3/2 X(5886) {(E+4F)$a$+2$aS A $}/6 1/3 X(5901) {(E+4F)$a$+4$aS A $}/8 1/2 X(6361) (2E-7F)$a$-2$aS A $ - 2 X(6684) {(2E-F)$a$+-$aS A $}/4-1/4 X(7967) {-3F$a$+4$aS A $}/3 4/3 X(7982) -(E-2F)$a$+3$aS A $ 3 X(7987) {2(E-2F)$a$+$aS A $}/5 1/5 X(7988) {2(E+4F)$a$+$aS A $}/9 1/9 X(7989) {2(E+4F)$a$-$aS A $}/7-1/7 X(7991) 2(E-2F)$a$-3$aS A $ - 3

X(8148) {-3(E-2F)$a$+8$aS A $}/2 4 X(8227) {(E+4F)$a$+$aS A $}/5 1/5 X(9588) {2(2E-F)$a$-3$aS A $}/7-3/7 X(9589) -2(E-5F)$a$+3$aS A $ 3 X(9624) {(E+4F)$a$+3$aS A $}/7 3/7 X(9778) {3(E-3F)$a$-2$aS A $}/3-2/3 X(9780) {3(E+F)$a$-2$aS A $}/7-2/7 X(9812) {-(E-11F)$a$+2$aS A $}/3 2/3 X(9955) {(E+10F)$a$+2$aS A $}/8 1/4 X(9956) {3(E+2F)$a$-2$aS A $}/8-1/4 X(10164) {3(E-F)$a$-$aS A $}/6-1/6 X(10165) {(2E-F)$a$+$aS A $}/6 1/6 X(10171) {3(E+3F)$a$+$aS A $}/12 1/12 X(10172) {(4E+7F)$a$-$aS A $}/12-1/12 X(10175) {(2E+5F)$a$-$aS A $}/6-1/6 X(10222) {(-E+2F)$a$+6$aS A $}/4 3/2 X(10246) {(E-2F)$a$+4$aS A $}/6 2/3 X(10247) {-(E-2F)$a$+8$aS A $}/6 4/3 X(10248) {-E+23F)$a$+2$aS A $}/7 2/7 X(10283) {(E+4F)$a$+8$aS A $}/12 2/3 X(10595) {3F$a$+4$aS A $}/5 4/5 X(11224) {-2(E-2F)$a$+7$aS A $}/3 7/3 X(11230) {3(E+2F)$a$+2$aS A $}/12 1/6 X(11231) {(5E+2F)$a$-2$aS A $}/12-1/6 X(11278) {-3(E-2F)$a$+10$aS A $}/4 5/2 X(11362) {(2E-F)$a$-3$aS A $}/2-3/2 X(11522) {6F$a$+3$aS A $}/5 3/5 X(11531) -2(E-2F)$a$+5$aS A $ 5 X(12245) (2E-F)$a$-4$aS A $ - 4 X(12512) {3(E-3F)$a$-$aS A $}/4-1/4 X(12571) {(E+13F)$a$+$aS A $}/8 1/8 X(12645) {3(E+2F)$a$-8$aS A $}/2-4 X(12699) {-(E-8F)$a$+2$aS A $}/2 1 X(12702) {3(E-2F)$a$-4$aS A $}/2-2 X(13607) {-3F$a$+5$aS A $}/4 5/4

X(13464) {3F$a$+3$aS A $}/4 3/4 X(13624) {3(E-2F)$a$+2$aS A $}/8 1/4 X(15178) {(E-2F)$a$+6$aS A $}/8 3/4 X(15808) {3(E+F)$a$+5$aS A $}/14 5/14 X(16189) {-2(E-2F)$a$+9$aS A $}/5 9/5 X(16191) {-4(E-2F)$a$+17$aS A $}/9 17/9 X(16192) {4(E-2F)$a$-$aS A $}/7-1/7 X(16200) {-(E-2F)$a$+5$aS A $}/3 5/3 Table 5 Euler coordinates (xe(n), ye(n)) of X(n) in N=E-class X(n) xe(n)*e ye(n)*e/(s B -S C )(S C -S A )(S A -S B ) X(51) {2(E+F)F+2S 2 }/3-1/3 X(52) {-(E 2-4F 2 )+4S 2 }/2-1 X(143) {-(E 2-4EF-8F 2 )+8S 2 }/8-1/2 X(185) -2(2E-F)F+2S 2-1 X(373) {2(E+F) 2 +2S 2 }/9-1/9 X(389) {-(E-2F)F+2S 2 }/2-1/2 X(568) {-(E 2 +2EF-8F 2 )+8S 2 }/6-2/3 X(1216) {(3E 2 +2EF-4F 2 )-4S 2 }/4 1/2 X(2979) {(3E 2 -EF-4F 2 )-4S 2 }/3 2/3 X(3060) {-(E 2-3EF-4F 2 )+4S 2 }/3-2/3 X(3567) {(E+4F)F+4S 2 }/5-2/5 X(3819) {(3E 2 +EF-2F 2 )-2S 2 }/6 1/6 X(3917) {2(E 2 -F 2 )-3S 2 }/3 1/3 X(5446) {-(E 2-6EF-4F 2 )+4S 2 }/4-1/2 X(5447) {(5E 2-2EF-4F 2 )-4S 2 }/8 1/4 X(5462) {(E 2 +2EF+5F 2 )+4S 2 }/8-1/4 X(5562) (E 2 +2EF-2F 2 )-2S 2 1 X(5640) {(E 2 +5EF+4F 2 )+4S 2 }/9-2/9 X(5650) {(4E 2 +2EF-2F 2 )-2S 2 }/3 1/3 X(5876) {(3E 2 +15EF-8F 2 )-8S 2 }/4 1 X(5889) -(E 2 +3EF-4F 2 )+4S 2-2 X(5890) {-(5E-4F)F+4S 2 }/3-2/3

X(5891) {(3E 2 +8EF-4F 2 )-4S 2 }/6 1/3 X(5892) {(3E 2-2EF+4F 2 )+4S 2 }/12-1/6 X(5907) {(E 2 +5EF-2F 2 )-2S 2 }/2 1/2 X(5943) {(E 2 +3EF+2F 2 )+2S 2 }/6-1/6 X(5946) {(E 2 +8F 2 )+8S 2 }/12-1/3 X(6101) {(5E 2-8F 2 )-8S 2 }/4 1 X(6102) {-(E 2 +8EF-8F 2 )+8S 2 }/4-1 X(6146) -3(E+2F)F+2S 2-1 X(6241) -(11E-4F)F+4S 2-2 X(6243) {-(3E 2-2EF-8F 2 )+8S 2 }/2-2 X(6688) {(3E 2 +5EF+2F 2 )+2S 2 }/12-1/12 X(7998) {(5E 2 +EF-4F 2 )-4S 2 }/9 2/9 X(7999) {(4E 2 +3EF-4F 2 )-4S 2 }/7 2/7 X(9729) {(E 2-3EF+2F 2 )+2S 2 }/4-1/4 X(9730) {(E 2-4EF+4F 2 )+4S 2 }/6-1/3 X(9781) {(7E+4F)F+4S 2 }/7-2/7 X(10095) {(E 2 +12EF+8F 2 )+8S 2 }/16-1/4 X(10110) {(5E+2F)F+2S 2 }/4-1/4 X(10112) {-(E 2 +5EF+10F 2 )+6S 2 }/2-3/2 X(10116) {-(E 2 +18EF+20F 2 )+12S 2 }/4-3/2 X(10170) {(5E 2 +10EF-4F 2 )-4S 2 }/12 1/6 X(10219) {(7E 2 +9EF+2F 2 )+2S 2 }/24-1/24 X(10263) {-(3E 2-8EF-8F 2 )+8S 2 }/4-1 X(10574) {(E 2-7EF+4F 2 )+4S 2 }/5-2/5 X(10575) {(E 2-16EF+4F 2 )+4S 2 }/2-1 X(10625) {(3E 2-4EF-4F 2 )-4S 2 }/2 1 X(10627) {(7E 2-4EF-8F 2 )-8S 2 }/8 1/2 X(11002) {-(E 2-7EF-8F 2 )+8S 2 }/9-4/9 X(11017) {(9E 2 +52EF-8F 2 )-8S 2 }/32 1/8 X(11225) {-(E 2 +7EF+6F 2 )+10S 2 }/5-1 X(11245) {-4(E+F)F+4S 2 }/3-2/3 X(11264) {-(3E 2 +28EF+40F 2 )+24S 2 }/8-3/2 X(11381) 2(5E-F)F-2S 2 1 X(11412) (2E 2 +EF-4F 2 )-4S 2 2 X(11439) {(E 2 +21EF-4F 2 )-4S 2 }/5 2/5

X(11444) {(3E 2 +5EF-4F 2 )-4S 2 }/5 2/5 X(11451) {(3E 2 +7EF+4F 2 )+4S 2 }/15-2/15 X(11455) {(23E-4F)F-4S 2 }/3 2/3 X(11459) {(2E 2 +7EF-4F 2 )-4S 2 }/3 2/3 X(11465) {(4E 2 +5EF+4F 2 )+4S 2 }/17-2/17 X(11565) {(3E 2-44EF-104F 2 )+24S 2 }/16-3/4 X(11591) {(5E 2 +12EF-8F 2 )-8S 2 }/8 1/2 X(11592) {(17E 2-12EF-8F 2 )-8S 2 }/32 1/8 X(11695) {(2E 2 +EF+2F 2 )+2S 2 }/8-1/8 X(11754) {(4E+9F)F+2S 2 }/4-1/4 X(11750) {(E 2-12EF-28F 2 )+4S 2 }/2-1 X(11793) {(2E 2 +3EF-2F 2 )-2S 2 }/4 1/4 X(12002) {-(3E 2-30EF-12F 2 )+12S 2 }/16-3/8 X(12006) {(3E 2-4EF+8F 2 )+8S 2 }/16-1/4 X(12022) {-3(E+4F)F+4S 2 }/3-2/3 X(12024) {-3(2E+5F)F+5S 2 }/3-5/6 X(12045) {(11E 2 +13EF+2F 2 )+2S 2 }/36-1/36 X(12046) {(15E 2 +44EF+8F 2 )+8S 2 }/64-1/16 X(12111) (E 2 +9EF-4F 2 )-4S 2 2 X(12134) {(E 2 +10EF+12F 2 )-4S 2 }/2 1 X(12162) {(E 2 +12EF-4F 2 )-4S 2 }/2 1 X(12241) -3F 2 +S 2-1/2 X(12278) (E 2 +5EF+28F 2 )-4S 2 2 X(12279) (E 2-19EF+4F 2 )+4S 2-2 X(12289) -(7E-32F)F+4S 2-2 X(12290) (17E-4F)F-4S 2 2 X(12370) {-(E 2 +4EF+24F 2 )+8S 2 }/4-1 X(12897) {-(E 2-10EF+28F 2 )+4S 2 }/4-1/2 X(13142) {-(E 2 -EF+4F 2 )+4S 2 }/2-1 X(13292) {-(E 2 +6EF-4F 2 )+8S 2 }/4-1 X(13340) {(7E 2-10EF-8F 2 )-8S 2 }/6 2/3 X(13348) {(3E 2-5EF-2F 2 )-2S 2 }/4 1/4 X(13363) {(5E 2 +4EF+8F 2 )+8S 2 }/24-1/6 X(13364) {(3E 2 +20EF+8F 2 )+8S 2 }/24-1/6 X(13382) {-3(3E-2F)F+6S 2 }/4-3/4

X(13403) {(E-14F)F+2S 2 }/2-1/2 X(13419) {(11E+14F)F-2S 2 }/2 1/2 X(13421) {-(9E 2-12EF-24F 2 )+24S 2 }/8-3/2 X(13451) {-(E 2-24EF-16F 2 )+16S 2 }/24-1/3 X(13470) {(3E 2-20EF-56F 2 )+8S 2 }/8-1/2 X(13474) {(13E-2F)F-2S 2 }/2 1/2 X(13491) {(E 2-24EF+8F 2 )+8S 2 }/4-1 X(13568) -(E-5F)F+S 2-1/2 X(13570) {(E 2 +21EF+2F 2 )+2S 2 }/12-1/12 X(13598) {-(E 2-7EF-2F 2 )+2S 2 }/2-1/2 X(13630) {(E 2-12EF+8F 2 )+8S 2 }/8-1/2 X(14128) {(7E 2 +20EF-8F 2 )-8S 2 }/16 1/4 X(14449) {-(5E 2-8EF-16F 2 )+16S 2 }/12-1 X(14516) (E 2 +7EF+12F 2 )-4S 2 2 X(14531) -2(E 2 +EF-3F 2 )+6S 2-3 X(14641) {(3E 2-26EF+4F 2 )+4S 2 }/4-1/2 X(14831) {-(E 2 +4EF-6F 2 )+6S 2 }/3-1 X(14845) {(3E 2 +16EF+4F 2 )+4S 2 }/18-1/9 X(14855) {(3E 2-20EF+4F 2 )+4S 2 }/6-1/3 X(15012) {(E 2-5EF+6F 2 )+6S 2 }/8-3/8 X(15028) {(3E 2 +EF+4F 2 )+4S 2 }/13-2/13 X(15030) {(E 2 +8EF-2F 2 )-2S 2 }/3 1/3 X(15043) (E 2 -EF+4F 2 )+4S 2-2/7 X(15060) {(5E 2 +24EF-8F 2 )-8S 2 }/12 1/3 X(15067) {(7E 2 +8EF-8F 2 )-8S 2 }/12-1/3 X(15072) {(E 2-13EF+4F 2 )+4S 2 }/3-2/3 X(15305) {(E 2 +15EF-4F 2 )-4S 2 }/3 2/3 X(15606) {(4E 2 +EF-6F 2 )-6S 2 }/4 3/4 X(15644) {(2E 2-3EF-2F 2 )-2S 2 }/2 1/2 X(15807) {(E 2 +20EF-56F 2 )+8S 2 }/16-1/4 X(16194) {(E 2 +24EF-4F 2 )-4S 2 }/6 1/3 X(16226) {(E 2-2EF+6F 2 )+6S 2 }/9-1/3 Table 6 Euler coordinates (xe(n), ye(n)) of X(n) in N =F -class

X(n) xe(n)*f ye(n)*f/{(s B -S C )(S C -S A )(S A -S B )} X(64) 2(2E-F)F-S 2 1/2 X(68) {(E+2F)(E+4F)-4S 2 }/4 1/2 X(154) {-2(E+F)F+S 2 }/3-1/6 X(156) {-(E 2 +12EF+8F 2 )+8S 2 }/16-1/4 X(1147) {-(E 2 +2EF+4F 2 )+4S 2 }/8-1/4 X(1498) -3EF+S 2-1/2 X(1853) {4(E+F)F-S 2 }/3 1/6 X(2883) {-3(E-F)F+S 2 }/2-1/4 X(3167) {-(E+F)(E+2F)+4S 2 }/6-1/3 X(3357) {3(3E-2F)F-2S 2 }/4 1/4 X(5448) {-(E 2 +2EF-20F 2 )+4S 2 }/16-1/8 X(5449) {(E 2 +10EF+12F 2 )-4S 2 }/16 1/8 X(5654) {-(E+4F)(E-2F)+4S 2 }/12-1/6 X(5656) {-3(2E-F)F+2S 2 }/3-1/3 X(5878) {-(7E-8F)F+2S 2 }/2-1/2 X(5893) {-6(E-3F)F+2S 2 }/8-1/8 X(5894) {(5E-7F)F-S 2 }/2 1/4 X(5895) -4(E-2F)F+S 2-1/2 X(5925) 5(E-2F)F-S 2 1/2 X(5944) {-(3E 2 +16EF+28F 2 )+16S 2 }/28-2/7 X(6193) {-(E 2 +4EF+6F 2 )+4S 2 }/2-1 X(6225) -(7E-5F)F+2S 2-1 X(6247) {(4E+F)F-S 2 }/2 1/4 X(6696) {(5E-F)F-S 2 }/4 1/8 X(6759) {-(5E+2F)F+2S 2 }/4-1/4 X(7689) {(E 2 +10EF-12F 2 )-4S 2 }/8 1/4 X(8567) {6(E-F)F-S 2 }/5 1/10 X(9820) {-(E 2-4F 2 )+4S 2 }/16 1/8 X(9833) {-(5E+8F)F+2S 2 }/2-1/2 X(9927) {(E 2 +6EF+20F 2 )-4S 2 }/8 1/4 X(9936) {-(3E 2 +14EF+8F 2 )+12S 2 }/4-3/2 X(10182) {3(E-2F)F+2S 2 }/24-1/24 X(10192) {-(E+F)F+S 2 }/6-1/12

X(10193) {(17E-10F)F-2S 2 }/24 1/24 X(10282) {-3(E+2F)F+2S 2 }/8-1/8 X(10606) {(5E-4F)F-S 2 }/3 1/6 X(11202) {-(E+10F)F+2S 2 }/12-1/12 X(11204) {(13E-14F)F-2S 2 }/12 1/12 X(11206) {-5(E+F)F+2S 2 }/3-1/3 X(11411) {(E 2 +6EF+2F 2 )-4S 2 }/2 1 X(12038) {-(E 2-2EF+12F 2 )+4S 2 }/16-1/8 X(12118) {-(E 2 +2EF+16F 2 )+4S 2 }/4-1/2 X(12163) {(E 2 +8EF-8F 2 )-4S 2 }/4 1/2 X(12164) {-(E 2 +5EF-2F 2 )+4S 2 }/2-1 X(12250) (8E-7F)F-2S 2 1 X(12293) {(E 2 +4EF+24F 2 )-4S 2 }/4 1/2 X(12315) {-(13E-2F)F+4S 2 }/2-1 X(12324) (7E+F)F-2S 2 1 X(12359) {(7E+4F)F-2S 2 }/8 1/4 X(12429) {(E 2 +5EF+10F 2 )-4S 2 }/2 1 X(13093) {3(5E-2F)F-4S 2 }/2 1 X(13346) {-(E 2-3EF+2F 2 )+2S 2 }/4-1/4 X(13561) {(E+28F)E-8S 2 }/32 1/8 X(14216) {(7E+4F)F-2S 2 }/2 1/2 X(14530) {3(3E+2F)F+4S 2 }/10-1/5 X(14852) {(E+4F) 2-4S 2 }/12 1/6 X(14862) {-3(5E-2F)F+6S 2 }/16-3/16 X(14864) {3(7E+6F)F-6S 2 }/8 3/8 X(16252) {-(2E-F)F+S 2 }/4-1/8 Table 7 Euler coordinates (xe(n), ye(n)) of X(n) in N = (E-8F)-class X(n) xe(n)*(e-8f) ye(n)*(e-8f)/{(s B -S C )(S C -S A )(S A -S B )} X(74) -3(5E-4F)F+4S 2-2 X(110) (E 2 +5EF+4F 2 )-4S 2 2 X(113) {(E 2 +8EF-20F 2 )-4S 2 }/2 1 X(125) -6(E+F)F+2S 2-1

X(146) (E 2 +23EF-32F 2 )-8S 2 4 X(265) {-(E 2 +14EF+40F 2 )+8S 2 }/2-2 X(323) 3(E+F)E-8S 2 4 X(399) {3(E+10F)F-16S 2 }/2 4 X(1495) 6(E+F)F-2S 2 1 X(1511) {3(E 2 +8F 2 )-8S 2 }/4 1 X(1514) 9(E-2F)F-2S 2 1 X(1531) (E 2 +2EF-26F 2 )-2S 2 1 X(1533) -(E 2-16EF+10F 2 )-2S 2 1 X(1539) {(E 2 +32EF-104F 2 )-8S 2 }/4 1 X(1568) (E 2-8F 2 )-2S 2 1 X(3292) 2(E 2 +2EF+F 2 )-6S 2 3 X(3448) -(E 2 +17EF+16F 2 )+8S 2-4 X(3580) -(E 2 +5EF+4F 2 )+4S 2-2 X(3581) {-3(E 2 +2EF-8F 2 )+8S 2 }/2-2 X(5609) {(5E 2 +40EF+8F 2 )-24S 2 }/4 3 X(5642) {(2E 2 +4EF+2F 2 )-6S 2 }/3 1 X(5655) {(5E 2 +46EF-40F 2 )-24S 2 }/6 2 X(5972) {(E 2 -EF-2F 2 )-2S 2 }/2 1/2 X(6053) {(4E 2 +19EF-10F 2 )-10S 2 }/2 5/2 X(6699) {(E 2-22EF+4F 2 )+4S 2 }/4-1/2 X(6723) {(E 2-13EF-14F 2 )+2S 2 }/4-1/4 X(7687) {-3(E+10F)F+2S 2 }/2-1/2 X(7728) {(E 2 +26EF-56F 2 )-8S 2 }/2 2 X(9140) {-(E 2 +29EF+28F 2 )+12S 2 }/3-2 X(9143) {(5E 2 +37EF+32F 2 )-24S 2 }/3 4 X(10113) {(7E 2 +8EF-80F 2 )+8S 2 }/4-1 X(10264) {-(E 2 +44EF+16F 2 )+16S 2 }/4-2 X(10272) {(5E 2 +16EF-16F 2 )-16S 2 }/8 1 X(10540) {(E 2 +18EF+8F 2 )-8S 2 }/2 2 X(10564) {(3E 2-12EF+12F 2 )-4S 2 }/2 1 X(10620) {-(E 2 +50EF-32F 2 )+16S 2 }/2-4 X(10706) {(2E 2 +31EF-52F 2 )-12S 2 }/3 2 X(10721) 3(7E-20F)F-4S 2 2 X(10733) -(E 2 -EF+52F 2 )+4S 2-2

X(10990) -6(4E-5F)F+6S 2-3 X(11064) (E 2 -EF-2F 2 )-2S 2 1 X(11693) {(13E 2 +8EF+76F 2 )-36S 2 }/18 1 X(11694) {(17E 2 +16EF+80F 2 )-48S 2 }/24 1 X(11801) {-(E 2 +32EF+112F 2 )+16S 2 }/8-1 X(12041) {(E 2-40EF+40F 2 )+8S 2 }/4-1 X(12112) 27EF-8S 2 4 X(12121) {(3E 2-6EF+72F 2 )-8S 2 }/2 2 X(12244) -3(11E-16F)F+8S 2-4 X(12295) {-(E 2-4EF+76F 2 )+4S 2 }/2-1 X(12308) (5E 2 +70EF-16F 2 )-32S 2 8 X(12317) -(2E 2 +37EF+8F 2 )+16S 2-8 X(12383) (2E 2 +7EF+32F 2 )-8S 2 4 X(12900) {{3(E 2-2EF-12F 2 )-4S 2 }/8 1/4 X(12902) {-(3E 2 +18EF+96F 2 )+16S 2 }/2-4 X(13202) 6(2E-7F)F-2S 2 1 X(13392) {(11E 2 +16EF+32F 2 )-32S 2 }/16 1 X(13393) {-3(3E 2 +80EF+32F 2 )+96S 2 }/16-3 X(13399) -2(10E+F)F+6S 2-3 X(13455) (E 2-23EF+12F 2 )+4S 2-2 X(13857) {2(2E 2-5EF-7F 2 )-6S 2 }/3 1 X(14094) (2E 2 +25EF-4F 2 )-12S 2 6 X(14156) {(3E 2-10EF-4F 2 )-4S 2 }/4 1/2 X(14157) (13E+4F)F-4S 2 2 X(14643) {(3E 2 +6EF-24F 2 )-8S 2 }/6 2/3 X(14644) {-9(E+4F)F+4S 2 }/3-2/3 X(14677) {(E 2-76EF+112F 2 )+16S 2 }/4-2 X(14683) 3(E 2 +9EF+8F 2 )-16S 2 8 X(15021) {(E 2-55EF+52F 2 )+12S 2 }/5-6/5 X(15027) {-(E 2 +62EF+88F 2 )+24S 2 }/10-6/5 X(15035) {(2E 2-5EF+20F 2 )-4S 2 }/3 2/3 X(15041) {(17E 2-38EF+80F 2 )-16S 2 }/6 4/3 X(15044) {-(E 2 +17EF+124F 2 )+12S 2 }/7-6/7 X(15051) {3(E 2-5EF+12F 2 )-4S 2 }/5 2/5 X(15054) -(E 2 +35EF-20F 2 )+12S 2-6

X(15055) (E 2-25EF+28F 2 )+4S 2-2 X(15063) (E 2 +14EF-14F 2 )-6S 2 3 X(15088) {3(E 2-12EF-40F 2 )+16S 2 }/16-1/4 X(15107) -3(E 2-3EF-4F 2 )+4S 2-2 X(15360) {-(5E 2 +EF-4F 2 )+12S 2 }/3-2 X(15361) {-(7E 2 +32EF-56F 2 )+24S 2 }/12-1 X(15362) {-(7E 2 +14EF+64F 2 )+24S 2 }/18-2/3 X(15448) 3(E+F)F-S 2 1/2 X(16003) {-(E 2 +32EF+4F 2 )+12S 2 }/2-3 X(16111) {(E 2-28EF+52F 2 )+4S 2 }/4-1 X(16163) (E 2-4EF+22F 2 )-2S 2 }/2 1 Table 8 Euler coordinates (xe(n), ye(n)) of X(n) in N = (E+F)-class X(n) xe(n)*(e+f) ye(n)*(e+f)/{(s B -S C )(S C -S A )(S A -S B )} X(6) S 2-1 X(69) (E+F) 2-2S 2 1 X(141) (E+F) 2 -S 2 1/4 X(182) {(E+F)(E-2F)+2S 2 }/4-1 X(193) -(E+F) 2 +4S 2-2 X(575) {(E+F)(E-2F)+6S 2 }/8-3/8 X(576) {-(E+F)(E-2F)+6S 2 }/4-3/4 X(597) {(E+F) 2 +3S 2 }/6-1/4 X(599) {2(E+F) 2-3S 2 }/3 1/2 X(1350) (E+F)(E-2F)-S 2 1/2 X(1351) {-(E+F)(E-2F)+4S 2 }/2-1 X(1352) {(E+F)(E+4F)-2S 2 }/2 1/2 X(1353) {-(E+F)(E+4F)+8S 2 }/4-1 X(1992) {-(E+F) 2 +6S 2 }/3-1 X(3018) {(E+F)(E+10F)-2S 2 }/4 1/4 X(3098) {3(E+F)(E-2F)-2S 2 }/4 1/4 X(3589) {(E+F) 2 +S 2 }/4-1/8 X(3618) {(E+F) 2 +2S 2 }/5-1/5 X(3619) {3(E+F) 2-2S 2 }/7 1/7

X(3620) {3(E+F) 2-4S 2 }/5 2/5 X(3629) {-(E+F) 2 +10S 2 }/4-5/4 X(3630) {3(E+F) 2-7S 2 }/2 7/4 X(3631) {3(E+F) 2-5S 2 }/4 5/8 X(3763) {2(E+F) 2 -S 2 }/5 1/10 X(3818) {(E+F)(E+4F)-2S 2 }/7 1/7 X(5032) {-(E+F) 2 +12S 2 }/9-2/3 X(5050) {(E+F)(E-2F)+4S 2 }/6-1/3 X(5085) {(E+F)(E-2F)+S 2 }/3-1/6 X(5092) {3(E+F)(E-2F)+2S 2 }/8-1/8 X(5093) {-(E+F)(E-2F)+8S 2 }/6-2/3 X(5097) {-(E+F)(E-2F)+10S 2 }/8-5/8 X(5102) {-(E+F)(E-2F)+5S 2 }/3-5/6 X(5476) {(E+F)(E+10F)+6S 2 }/12-1/4 X(5480) {3(E+F)F+S 2 }/2-1/4 X(5921) (E+F)(E+7F)-4S 2 2 X(6144) -2(E+F) 2 +7S 2-7/2 X(6329) {(E+F) 2 +5S 2 }/8-5/16 X(6776) -3(E+F)F+2S 2-1 X(8550) {-3(E+F)F+3S 2 }/2-3/4 X(8584) {-(E+F) 2 +9S 2 }/6-3/4 X(10168) {(E+F)(7E-2F)+6S 2 }/24-1/8 X(10516) {(E+F)(E+4F)-S 2 }/3 1/6 X(10519) {(E+F)(2E-F)-2S 2 }/3 1/3 X(10541) {2(E+F)(E-2F)+3S 2 }/7-3/14 X(11008) -3(E+F) 2 +10S 2-5 X(11160) {5(E+F) 2-12S 2 }/3 2 X(11178) {(E+F)(5E+14F)-6S 2 }/12 1/4 X(11179) {(E+F)(E-8F)+6S 2 }/6-1/2 X(11180) {(E+F)(2E+11F)-6S 2 }/3 1 X(11477) -(E+F)(E-2F)+3S 2-3/2 X(11482) {-(E+F)(E-2F)+12S 2 }/10-3/5 X(11898) {3(E+F)(E+4F)-8S 2 }/2 2 X(12007) {-3(E+F)F+5S 2 }/4-5/8 X(12017) {3(E+F))(E-2F)+4S 2 }/10-1/5

X(14561) {(E+F)(E+4F)+2S 2 }/6-1/6 X(14810) {5(E+F)(E-2F)-2S 2 }/8 1/8 X(14848) {(E+F)(E+10F)+12S 2 }/18-1/3 X(14853) {3(E+F)F+2S 2 }/3-1/3 X(14912) {-3(E+F)F+4S 2 }/3-2/3 X(14927) (E+F)(E-11F)+2S 2-1 X(15069) (E+F)(E-2F)-3S 2 3/2 X(15516) {(E+F)(E-2F)+14S 2 }/16-7/16 X(15520) {-(E+F)(E-2F)+14S 2 }/12-7/12 X(15533) {4(E+F) 2-9S 2 }/3 3/2 X(15534) {7(E+F) 2-9S 2 }/3 3/2 Table 9 Euler coordinates (xe(n), ye(n)) of X(n) in N = (E+F) 2 -S 2 -class X(n) xe(n)*n ye(n)*n/{(e+f)(s B -S C )(S C -S A )(S A -S B )} X(32) (E+4F)S 2 /2-1/2 X(315) (E+F) 3 -(2E+5F)S 2 1 X(626) {2(E+F) 3-3(E+2F)S 2 }/4 1/4 X(1975) (E+F) 3-3(E-F)S 2 1 X(3933) {2(E+F) 3 -(5E+2F)S 2 }/2 1 X(5254) (E-2F)S 2-1/2 X(5305) 3ES 2 /4-1/2 X(5306) (E+4F)S 2 /6-1/2 X(5309) (5E-4F)S 2 /6-1/2 X(5346) (7E+4F)S 2 /10-1/2 X(6680) {2(E+F) 3 -(E-2F)S 2 }/8-1/8 X(7748) 3(E-4F)S 2 /2-1/2 X(7754) -(E+F) 3 +(4E+F)S 2-2 X(7788) {3(E+F) 3-7(E+F)S 2 }/3 1 X(7789) {(E+F) 3 -(2E-F)S 2 }/2 1/4 X(7801) {4(E+F) 3-9ES 2 }/6 1/2 X(7805) {-2(E+F) 3 +3(3E+2F)S 2 }/4-5/4 X(7816) {2(E+F) 3-5(E-2F)S 2 }/4 1/4 X(7817) {(E+F) 3-3FS 2 }/6-1/4

X(7818) {4(E+F) 3 -(7E+16F)S 2 }/6 1/2 X(7842) {2(E+F) 3 -(E+22F)S 2 }/4 1/4 X(7851) {(E+F) 3 +(E-5F)S 2 }/5-1/5 X(7855) {4(E+F) 3 -(11E+8F)S 2 }/2 5/2 X(7861) {2(E+F) 3 +(E-14F)S 2 }/8-1/8 X(7867) {4(E+F) 3 -(5E+8F)S 2 }/10 1/10 X(7880) {6(E+F) 3 -(11E+2F)S 2 }/12 1/4 X(7881) {3(E+F) 3-3(2E+F)S 2 }/5 2/5 X(7882) {6(E+F) 3-3(5E+6F)S 2 }/4 7/4 X(7895) {6(E+F) 3 -(13E+10F)S 2 }/8 5/8 X(9753) {3(E+F) 2 F+(E+F)S 2 }/3-1/3 X(11648) (7E-20F)S 2 /6-1/2 X(14614) {-(E+F) 3 +5(E+F)S 2 }/3-1 X(14880) {(E+F) 3 +3ES 2 }/3-1/2 Table 10 Euler coordinates (xe(n), ye(n)) of X(n) in N = (E+F) 2 +S 2 -class X(n) xe(n)*n ye(n)*n/{(e+f)(s B -S C )(S C -S A )(S A -S B )} X(39) 3ES 2 /2-1/2 X(76) (E+F) 3 -(2E-F)S 2 1 X(194) -(E+F) 3 +(5E-F)S 2-2 X(262) (E+F) 2 F+(E+F)S 2-1/3 X(598) {(E+F) 3 +3(2E+11F)S 2 }/9-1/3 X(3095) {-(E+F) 3 +3(E+F) 2 F+(5E+2F)S 2 }/2-1 X(3934) {2(E+F) 3 -(E-2F)S 2 }/4 1/4 X(5188) {2(E+F) 3-6(E+F) 2 F-(E+4F)S 2 }/2 1/2 X(6194) (E+F) 3-2(E+F) 2 F-(E+F)S 2 2/3 X(6248) {(E+F) 3 +3(E+F) 2 F-2(E-2F)S 2 }/2 1/2 X(6683) {2(E+F) 3 +(5E+2F)S 2 }/8-1/8 X(7697) {(E+F) 3 +(E+F) 2 F-(E-2F)S 2 }/2 1/3 X(7709) -(E+F) 2 F+(2E+F)S 2-2/3 X(7745) (E+4F)S 2-1/2 X(7747) (E+16F)S 2 /2-1/2 X(7750) (E+F) 3 -(E+7F)S 2 1

X(7753) (7E+16F)S 2 /6-1/2 X(7756) (5E-16F)S 2 /2-1/2 X(7757) {-(E+F) 3 +(8E-F)S 2 }/3-1 X(7762) -(E+F) 3 +(4E+7F)S 2-2 X(7767) {2(E+F) 3-3(E+2F)S 2 }/2 1 X(7786) {(E+F) 3 +(4E+F)S 2 }/5-1/5 X(7802) 2(E+F) 3-15FS 2 1 X(7810) {4(E+F) 3-3(E+4F)S 2 }/6 1/2 X(7811) {3(E+F) 3 -(4E+13F)S 2 }/3 1 X(7812) {-(E+F) 3 +3(2E+5F)S 2 }/3-1 X(7823) -(E+F) 3 +3(E+5F)S 2-2 X(7826) {4(E+F) 3-3(3E+4F)S 2 }/2 5/2 X(7830) {2(E+F) 3 +(E-14F)S 2 }/4 1/4 X(7837) {-3(E+F) 3 +13(E+F)S 2 }/3-2 X(7838) {-2(E+F) 3 +(11E+14F)S 2 }/4-5/4 X(7877) -3(E+F) 3 +(10E+13F)S 2-5 X(7890) {-4(E+F) 3 +3(5E+4F)S 2 }/2-7/2 X(7893) 3(E+F) 3 -(7E+13F)S 2 4 X(7904) {3(E+F) 3 -(E+13F)S 2 }/5 2/5 X(7921) {-(E+F) 3 +3(3E+5F)S 2 }/5-4/5 X(9300) 4(E+F)S 2 /3-1/2 X(9466) {4(E+F) 3 -(5E-4F)S 2 }/6 1/2 X(9774) {4(E+F) 3-27(E+F) 2 F+9(E+F)S 2 }/9-1/3 X(9821) {3(E+F) 3-9(E+F) 2 F-3(E+2F)S 2 }/2 1 X(10033) {2(E+F) 3 +27(E+F) 2 F-3(E+F)S 2 }/9 1/3 X(11055) {-5(E+F) 3 +(22E-5F)S 2 }/3-3 X(11057) {3(E+F) 3 -(2E+29F)S 2 }/3 1 X(11171) {(E+F) 3 -(E+F) 2 F+(7E-2F)S 2 }/6-1/3 X(11257) -3(E+F) 2 F+3(E-F)S 2-1 X(11272) {(E+F) 3 +3(E+F) 2 F+(7E+4F)S 2 }/8-1/4 X(13108) {3(E+F) 3 +3(E+F) 2 F-3(3E-2F)S 2 }/2 2 X(13334) {(E+F) 3-3(E+F) 2 F+2(2E-F)S 2 }/4-1/4 X(14458) {27(E+F) 2 F-5(E+F)S 2 }/3 1 X(14537) (5E+32F)S 2 /6-1/2 X(14711) {8(E+F) 3 -(19E-8F)S 2 }/6 3/2

X(14762) {10(E+F) 3 +3(5E+14F)S 2 }/36-1/12 X(14881) {-(E+F) 3 +9(E+F) 2 F+(5E+8F)S 2 }/4-1/2 X(14976) {5(E+F) 3 -(5E+59F)S 2 }/3 2 X(15810) {8(E+F) 3 +3(E-8F)S 2 }/18 1/6 X(15819) {(E+F) 3 -(E+F) 2 F}/2 1/6 Table 11 Euler coordinates (xe(n), ye(n)) of X(n) in N = (E+F) 2-3S 2 -class X(n) xe(n)*n ye(n)*n/{(e+f)(s B -S C )(S C -S A )(S A -S B )} X(98) -3(E+F) 2 F+(E+F)S 2-1 X(99) (E+F) 3 -(4E-5F)S 2 1 X(114) {(E+F) 3 +3(E+F) 2 F-4(E+F)S 2 }/2 1/2 X(115) (E-8F)S 2 /2-1/2 X(147) (E+F) 3 +6(E+F) 2 F-5(E+F)S 2 2 X(148) -(E+F) 3 +(5E-13F)S 2-2 X(187) -(E-8F)S 2 /2-1/2 X(230) 0-1/2 X(316) (E+F) 3 -(2E+11F)S 2 1 X(325) (E+F) 3-3(E+F)S 2 1 X(385) -(E+F) 3 +3(E+F)S 2-2 X(620) {2(E+F) 3 -(7E-2F)S 2 }/4 1/4 X(625) {2(E+F) 3 -(5E+14F)S 2 }/4 1/4 X(671) {-(E+F) 3 +3(2E-7F)S 2 }/3-1 X(2080) {-(E+F) 3 +3(E+F) 2 F+(E+10F)S 2 }/2-1 X(2482) {4(E+F) 3-3(5E-4F)S 2 }/6 1/2 X(5215) {4(E+F) 3-3(5E-4F)S 2 }/18-1/6 X(5461) {2(E+F) 3-3(E+10F)S 2 }/12-1/4 X(5984) -(E+F) 3-12(E+F) 2 F+7(E+F)S 2-4 X(6033) {(E+F) 3 +3(E+F) 2 F-(5E+14F)S 2 }/2 1 X(6036) {(E+F) 3-3(E+F) 2 F-2(E+F)S 2 }/4-1/4 X(6054) {2(E+F) 3 +9(E+F) 2 F-9(E+F)S 2 }/3 1 X(6055) {(E+F) 3-9(E+F) 2 F}/6-1/2 X(6321) {-(E+F) 3 +3(E+F) 2 F+(5E-22F)S 2 }/2-1 X(6390) {2(E+F) 3 -(7E-2F)S 2 }/2 1

X(6721) {3(E+F) 3 +3(E+F) 2 F-10(E+F)S 2 }/8 1/8 X(6722) {2(E+F) 3 -(5E+14F)S 2 }/8-1/8 X(6781) -3(E-8F)S 2 /2-1/2 X(7779) 3(E+F) 3-9(E+F)S 2 4 X(7799) {3(E+F) 3 -(10E+F)S 2 }/3 1 X(7809) {3(E+F) 3 -(8E+17F)S 2 }/3 1 X(7813) 4(E+F) 3 -(13E+4F)S 2 5/2 X(7840) {5(E+F) 3-15(E+F)S 2 }/3 2 X(7845) {4(E+F) 3 -(11E+20F)S 2 }/2 5/2 X(7925) {3(E+F) 3-9(E+F)S 2 }/5 2/5 X(8591) {5(E+F) 3-3(7E-11F)S 2 }/3 2 X(8596) {-7(E+F) 3 +3(11E-25F)S 2 }/3-4 X(8724) {5(E+F) 3 +9(E+F) 2 F-3(7E-2F)S 2 }/6 1 X(8859) {-(E+F) 3 +3(E+F)S 2 }/9-2/3 X(9166) {(E+F) 3-27FS 2 }/3-1/3 X(9167) {8(E+F) 3-27ES 2 }/18 1/6 X(9301) {-3(E+F) 3 +9(E+F) 2 F+(5E+14F)S 2 }/2-2 X(9862) -9(E+F) 2 F+(2E+11F)S 2-2 X(9880) {-(E+F) 3 +9(E+F) 2 F+6(E-8F)S 2 }/6-1/2 X(10150) {14(E+F) 3-3(13E+22F)S 2 }/36 1/12 X(10242) {(E+F) 3 +(E+F) 2 F-(E+22F)S 2 }/2 1/3 X(10722) 9(E+F) 2 F-(E+19F)S 2 1 X(10723) -(E+F) 3 +6(E+F) 2 F+(4E-23F)S 2-1 X(10991) {-12(E+F) 3 +3(E+4F)S 2 }/2-3/2 X(10992) {3(E+F) 3-3(E+F) 2 F-12(E-2F)S 2 }/2 3/2 X(11054) {-5(E+F) 3 +9(E-F)S 2 }/3-3 X(11177) {-(E+F) 3-18(E+F) 2 F+9(E+F)S 2 }/3-2 X(11623) {-6(E+F) 2 F+3(E-2F)S 2 }/4-3/4 X(11632) {-(E+F) 3-9(E+F) 2 F+9(E-2F)S 2 }/6-1 X(12042) {(E+F) 3-9(E+F) 2 F-(E-8F)S 2 }/4-1/2 X(12117) {4(E+F) 3-9(E+F) 2 F-3(5E-13F)S 2 }/3 1 X(12118) {-(E+F) 3-9(E+F) 2 F+(7E-2F)S 2 }/2-2 X(12243) {-2(E+F) 3-9(E+F) 2 F+3(4E-5F)S 2 }/3-2 X(12355) {-7(E+F) 3 +9(E+F) 2 F+3(11E-32F)S 2 }/6-2 X(13172) 2(E+F) 3-3(E+F) 2 F-(8E-19F)S 2 2

X(13188) {3(E+F) 3 +3(E+F) 2 F-(13E-14F)S 2 }/2 2 X(13449) {(E+F) 3 +3(E+F) 2 F-2(E+10F)S 2 }/2 1/2 X(14061) {(E+F) 3 -(2E+11F)S 2 }/5-1/5 X(14148) {6(E+F) 3-3(7E-2F)S 2 }/4 7/4 X(14568) {-(E+F) 3 +(4E-5F)S 2 }/3-1 X(14639) {3(E+F) 2 F+(E-17F)S 2 }/3-1/3 X(14651) {-3(E+F) 2 F+(2E-7F)S 2 }/3-2/3 X(14692) {5(E+F) 3 +21(E+F) 2 F-(25E-2F)S 2 }/2 5 X(14693) {(E+F) 3 +3(E+F) 2 F-(5E-4F)S 2 }/8-1/4 X(14712) -(E+F) 3 +(E+19F)S 2-2 X(14830) {(E+F) 3-27(E+F) 2 F+3(E+10F)S 2 }/6-1 X(14971) {4(E+F) 3-9(E+4F)S 2 }/18-1/6 X(14981) {2(E+F) 3 +6(E+F) 2 F -9ES 2 }/2 3/2 X(15300) {8(E+F) 3-3(11E-16F)S 2 }/6 3/2 X(15301) {6(E+F) 3 -(23E-22F)S 2 }/4 7/4 X(15561) {3(E+F) 3 +3(E+F) 2 F (11E+2F)S 2 }/6 1/3