Klein's quartic curve is a curve of genus 3 with the largest possible number of automorphisms for that genus. It is the first instance of equality for the Hurwitz bound. One model for it is the modular curve $X(7)$.
Family Information
| Genus: | $3$ |
| Dimension of the family: | $0$ |
Cover
| Quotient genus: | $0$ |
| Number of branch points: | $3$ |
| Signature: | $[ 0; 2, 3, 7 ]$ |
Group
| Name: | $\GL(3,2)$ |
| Identifier: | $[168,42]$ |
Conjugacy class(es) of Refined passports
| Refined passport label | Conjugacy classes |
|---|---|
| 3.168-42.0.2-3-7.1 | 2, 3, 5 |
| 3.168-42.0.2-3-7.2 | 2, 3, 6 |
Displaying the representative for the unique action up to topological equivalence.
3.168-42.0.2-3-7.1.1| (1,5) (2,9) (3,45) (4,88) (6,77) (7,104) (8,12) (10,17) (11,32) (13,154) (14,125) (15,19) (16,23) (18,39) (20,91) (21,160) (22,26) (24,66) (25,151) (27,84) (28,139) (29,33) (30,114) (31,38) (34,147) (35,48) (36,40) (37,163) (41,98) (42,69) (43,47) (44,51) (46,130) (49,146) (50,54) (52,59) (53,74) (55,112) (56,167) (57,61) (58,65) (60,81) (62,133) (63,118) (64,68) (67,109) (70,97) (71,75) (72,156) (73,80) (76,105) (78,82) (79,121) (83,140) (85,89) (86,93) (87,129) (90,161) (92,96) (94,101) (95,116) (99,103) (100,107) (102,123) (106,110) (108,150) (111,168) (113,117) (115,122) (119,132) (120,124) (126,153) (127,131) (128,135) (134,138) (136,143) (137,158) (141,145) (142,149) (144,165) (148,152) (155,159) (157,164) (162,166) | |
| (1,2,3) (4,5,6) (7,77,105) (8,9,10) (11,12,13) (14,154,126) (15,16,17) (18,19,20) (21,91,161) (22,23,24) (25,26,27) (28,84,140) (29,30,31) (32,33,34) (35,147,49) (36,37,38) (39,40,41) (42,98,70) (43,44,45) (46,47,48) (50,51,52) (53,54,55) (56,112,168) (57,58,59) (60,61,62) (63,133,119) (64,65,66) (67,68,69) (71,72,73) (74,75,76) (78,79,80) (81,82,83) (85,86,87) (88,89,90) (92,93,94) (95,96,97) (99,100,101) (102,103,104) (106,107,108) (109,110,111) (113,114,115) (116,117,118) (120,121,122) (123,124,125) (127,128,129) (130,131,132) (134,135,136) (137,138,139) (141,142,143) (144,145,146) (148,149,150) (151,152,153) (155,156,157) (158,159,160) (162,163,164) (165,166,167) | |
| (1,45,51,54,74,105,6) (2,5,88,161,20,15,10) (3,9,12,32,147,48,43) (4,77,104,99,94,86,89) (7,76,71,80,121,124,102) (8,17,23,26,151,126,13) (11,154,125,120,115,30,33) (14,153,148,108,100,103,123) (16,19,39,98,69,64,24) (18,91,160,155,164,37,40) (21,90,85,129,135,138,158) (22,66,58,61,81,140,27) (25,84,139,134,143,149,152) (28,83,78,73,156,159,137) (29,38,163,166,144,49,34) (31,114,117,95,70,41,36) (35,146,141,136,128,131,46) (42,97,92,101,107,110,67) (44,47,130,119,62,57,52) (50,59,65,68,109,168,55) (53,112,167,162,157,72,75) (56,111,106,150,142,145,165) (60,133,118,113,122,79,82) (63,132,127,87,93,96,116) |