Equation 677 Database

Translation-only Steiner-line magma at size 61, "near-Z_15-symmetric" slope pattern. Size 61, fully idempotent, RC. 183 size-5 sub-magmas (each = the F_5 affine line magma#e549b5f8). |Aut(M)| = 61 (only Z_61 translation, no multiplicative stabilizer). In the F_61 translation labeling, x*y = x + f(y - x) with f a non-linear permutation. Slope alpha(y) = f(y)/y takes 12 distinct values in F_61* on the 60 non-zero residues: 4 Phi_10 root slopes {3, 27, 41, 52}, each on a size-11 subset (44 elements total) 8 non-Phi_10 slopes {7, 10, 21, 30, 32, 35, 55, 59}, each on a size-2 subset (16 elements total) The 8 non-Phi slopes form 4 inverse pairs in F_61*: {7, 35}, {10, 55}, {21, 32}, {30, 59} (each pair multiplies to 1 mod 61). The 4 Phi_10 roots also pair as {3, 41} and {27, 52}. This magma is the unique Phi-sum=44 variant in the DB. It is the "closest approximation" to the fully Z_15-symmetric magma#0bcf3cca (which has Phi-sum = 60) among the translation-only Steiner-line magmas at size 61: of the 60 nonzero residues, 44 receive Phi_10-root slopes and only 16 deviate. The 16 deviating elements are sprinkled across F_61* as 8 inverse pairs. Compare with magma#2426bbe0, magma#ec9ec6f6 (the Phi-sum=28 pair: 4 Phi-slopes on size-7 + 8 non-Phi on size-4) - sharing the same 12-element slope value set. These four magmas (including this one) form a "tribe" using the unique 12-value slope set {3, 7, 10, 21, 27, 30, 32, 35, 41, 52, 55, 59} (= 4 Phi_10 roots + 8 inverse-paired extras). The other ~26 translation-only Steiner-line magmas at size 61 use larger, more chaotic slope-value sets (typically 20-34 distinct values). [text written by Claude]