Equation 677 Database

Translation-only Steiner-line magma at size 65 (Type B slope pattern - 2 magmas). Size 65 = 5 * 13, fully idempotent, RC. 208 size-5 sub-magmas on a cyclic S(2, 5, 65) design (each = the F_5 affine line magma#e549b5f8). |Aut(M)| = 65 (Z_65 translation only). In the Z_65 translation labeling, x*y = x + f(y - x) mod 65. Slope analysis on Z_65*: 12 distinct slope values {3, 15, 21, 22, 31, 32, 34, 35, 44, 45, 51, 63} (same set as in Type A and Type C - 4 inverse pairs + 1 self-inverse + 3 non-units), but with class sizes following the Type B pattern: 1 slope on a size-8 class 6 slopes on size-5 classes 5 slopes on size-2 classes Total: 8 + 30 + 10 = 48 = |Z_65*|. This Type B pattern is shared by exactly 2 size-65 magmas in the DB: this one and magma#2db4eae0. They share the slope value set but differ in the assignment. Compare: magma#eff27734 (Z_4-symmetric, |Aut|=260, 4 slope values, all classes size 12) Type A (5 magmas: 3 classes of size 6, 6 of size 4, 3 of size 2) this Type B (2 magmas: 1 class of size 8, 6 classes of size 5, 5 classes of size 2) Type C (1 magma magma#62917c53: 3 classes of size 9, 1 of size 8, 5 of size 2, 3 of size 1) [text written by Claude]