Equation 677 Database

Translation-only Steiner-line magma at size 65 (Type C slope pattern - unique variant). Size 65 = 5 * 13, fully idempotent, RC. 208 size-5 sub-magmas on a cyclic S(2, 5, 65) design. |Aut(M)| = 65 (Z_65 translation only). In the Z_65 translation labeling, slope analysis on Z_65* shows 12 distinct slope values (same value set {3, 15, 21, 22, 31, 32, 34, 35, 44, 45, 51, 63} as the other 7 non-Z_4-symmetric Steiner-line magmas at size 65), with the unique Type C class-size pattern: 3 slopes on size-9 classes (sum 27) 1 slope on a size-8 class 5 slopes on size-2 classes 3 slopes on size-1 classes Total: 27 + 8 + 10 + 3 = 48 = |Z_65*|. This magma is the unique Type C variant in the DB. The presence of 3 size-1 classes is distinctive - 3 specific units in Z_65* receive their own unique slope value, unmatched by any other unit. Compare: magma#eff27734 (Z_4-symmetric, |Aut|=260, 4 slope values, all classes size 12) Type A (5 magmas: magma#a09c4223, magma#86e94818, magma#4d60c34f, magma#bcf3ee89, magma#84076266 - 3 classes of size 6, 6 of size 4, 3 of size 2) Type B (2 magmas: magma#d8ba33b5, magma#2db4eae0 - 1 class of size 8, 6 of size 5, 5 of size 2) this Type C (this magma alone - 3 classes of size 9, 1 of size 8, 5 of size 2, 3 of size 1) [text written by Claude]