Equation 677 Database

Z_3-symmetric Steiner-line magma at size 61. Size 61, fully idempotent, RC. 183 size-5 sub-magmas (= F_5 affine line magma#e549b5f8). Every pair lies in exactly one block; each element in 15 blocks. |Aut(M)| = 183 = 3 * 61. Aut is Z_61 : Z_3: regular Z_61 acting by translation, and Z_3 = {1, 13, 47} (= the order-3 subgroup of F_61*, since 60 = 4*15 = 12*5 = 20*3, the order-3 subgroup is {1, 13, 47} where 13^3 = 47^3 = 1 mod 61). The Z_3 acts by multiplication. In the F_61 translation labeling (suggested reorder), the operation is x*y = x + f(y - x) where f is c-equivariant for c in {1, 13, 47}: f(13y) = 13 f(y) and f(47y) = 47 f(y). Equivalently, the slope function alpha(y) = f(y)/y is constant on the 20 Z_3-orbits of size 3 that partition F_61* = {1, ..., 60}. Slope analysis: alpha takes 16 distinct values across the 20 Z_3-orbits, all non-Phi_10 roots (i.e., NONE of {3, 27, 41, 52} appear as slopes!): alpha in {4, 6, 7, 8, 10, 17, 18, 19, 23, 29, 35, 40, 45, 46, 51, 55} 4 slopes occur on 2 Z_3-orbits each (size-6 slope classes) 12 slopes occur on 1 Z_3-orbit each (size-3 slope classes) This is structurally unusual: a translation-invariant magma whose slope function avoids all Phi_10 roots. The 16 chosen slopes satisfy a non-trivial Eq 677 functional equation jointly with the Z_3 equivariance. Compared to magma#0bcf3cca (the Z_15-symmetric variant): magma#0bcf3cca uses ONLY Phi_10-root slopes; this magma uses ONLY non-Phi_10-root slopes. The two represent opposite extremes of how the Z_61-translation structure can interact with multiplicative symmetry. Among the 36 size-61 Eq 677 magmas: 4 linear (|Aut|=3660), 1 with Z_15 stabilizer (magma#0bcf3cca, |Aut|=915), this one with Z_3 stabilizer (|Aut|=183), and 30 with only translation symmetry (|Aut|=61). [text written by Claude]