Equation 677 Database

Translation-only Steiner-line magma at size 65 (Type A slope pattern - 5 magmas). Size 65 = 5 * 13, fully idempotent, RC. 208 size-5 sub-magmas on a cyclic S(2, 5, 65) design. Each block = the F_5 affine line magma#e549b5f8. |Aut(M)| = 65 (only Z_65 translation, no multiplicative stabilizer). In the suggested reorder, Z_65 labeling makes translation x -> x+1 an automorphism. Operation: x*y = x + f(y - x) mod 65. Slope analysis on Z_65* (48 units): alpha takes 12 distinct values {3, 15, 21, 22, 31, 32, 34, 35, 44, 45, 51, 63}, with the following class sizes (Type A pattern): 3 slopes on size-6 classes (sum 18) 6 slopes on size-4 classes (sum 24) 3 slopes on size-2 classes (sum 6) Total: 18 + 24 + 6 = 48 = |Z_65*|. Three of the 12 slopes ({15, 35, 45}) are non-units mod 65 (multiples of 5); the other 9 are units and form 4 inverse pairs in Z_65* {3,22}, {21,31}, {32,63}, {34,44} plus the self-inverse 51 (= -14, 51^2 = 1 mod 65). This 12-value slope set is exactly the one obtained by taking all Z_65* elements that "could be Eq 677 multipliers" plus the 3 multiples of 5. Compare with magma#eff27734 (Z_4-symmetric) which uses only the 4-element subset {3, 22, 35, 51} as slopes - that subset is the Z_4 = <34> stabilizer multiplied by representative slope values. This (Type A) slope-class-size pattern is shared by 5 magmas at size 65: this one and magma#a09c4223, magma#86e94818, magma#bcf3ee89, magma#84076266. They differ in how the 12 slope values are assigned to specific Z_65*-subsets. Other Type B/C patterns (also using the same 12-value slope set) are: Type B (2 magmas: magma#d8ba33b5, magma#2db4eae0), Type C (1 magma: magma#62917c53). [text written by Claude]