Equation 677 Database

Translation-only Steiner-line magma on a cyclic S(2, 5, 61) design. Size 61, fully idempotent, RC. 183 = 61*60/(5*4) size-5 sub-magmas (each = the F_5 affine line magma#e549b5f8); every pair of distinct elements lies in exactly one block; each element is in 15 blocks. |Aut(M)| = 61 - only the Z_61 translation subgroup; no multiplicative stabilizer at any point. The 183 blocks split as 3 Aut-orbits of 61 each (cyclic BIBD with 3 base blocks under Z_61, since 183 = 3*61). In the suggested reorder, F_61 labeling makes translation x -> x+1 an automorphism. Operation: x*y = x + f(y - x) mod 61, where f is a non-linear permutation with f(0) = 0. Slope analysis (alpha(y) = f(y)/y on F_61*): 34 distinct slope values, far less symmetric than the size-41 analogs. Phi_10 roots in F_61 = {3, 27, 41, 52} (the primitive 10th roots of unity). The slope multiset: 4 Phi_10-root slopes appear on only 6 of the 60 nonzero elements (class sizes 1, 1, 2, 2) 30 non-Phi_10 slopes appear on the remaining 54 elements (class sizes ranging 1 to 5) The chaotic slope structure reflects the lack of higher symmetry: with only Aut = Z_61, the cyclic BIBD's difference family carries no extra multiplicative regularity, so the slope function is "as generic as possible" subject to the 677 constraint. Context: 36 size-61 Eq 677 magmas in the DB, split as 4 linear F_61 affine line magmas (one per Phi_10 root, |Aut| = 3660 = |AGL(1, 61)|), 1 magma#0bcf3cca with Z_15 multiplicative stabilizer (|Aut| = 915 = 15*61), 1 magma#05fbff2c with Z_3 multiplicative stabilizer (|Aut| = 183 = 3*61), 30 translation-only Steiner-line magmas (this one is one of them), |Aut| = 61. Compared to size 41: at size 41 there were only 8 Steiner-line magmas total (1 Z_5-symmetric + 7 translation-only), with a tidy 12-value slope set common to all. At size 61, 60 = 4*15 has more divisors than 40 = 8*5, giving Aut subgroups of orders 1, 3, 15, 60 (vs. 1, 5, 40 at size 41), hence many more variant types and a richer slope landscape. [text written by Claude]