Equation 677 Database

Size 61: 36 distinct Eq 677 magmas in the DB, all fully idempotent and right-cancellative. 61 is prime with 61 ≡ 1 (mod 10), so F_61 admits primitive 10th roots of unity. The four roots of the cyclotomic polynomial Phi_10(x) = x^4 - x^3 + x^2 - x + 1 in F_61 are {3, 27, 41, 52} (each of multiplicative order 10). The family splits into two structural classes plus a richer sub-classification by automorphism size: **4 Linear magmas** (F_61 affine line, x*y = x + alpha(y - x) = (1-alpha)x + alpha*y mod 61, one per Phi_10 root): magma#f2d8c865 (alpha = 52, equivalently x ◇ y = 10x + 52y) magma#d5df0d93 (alpha = 41, equivalently x ◇ y = 21x + 41y) magma#82a4911c (alpha = 27, equivalently x ◇ y = 35x + 27y) magma#e70931b6 (alpha = 3, equivalently x ◇ y = 59x + 3y) Each has |Aut| = 3660 = 60*61 = |AGL(1, 61)|. Zero size-5 sub-magmas (the F_61 line has no proper non-trivial sub-magmas). **32 Steiner-line magmas** on cyclic S(2, 5, 61) designs. Each has 183 = 61*60/(5*4) size-5 sub-magmas, each block is the F_5 affine line magma#e549b5f8, and every pair of distinct elements lies in exactly one block. They split by |Aut|: * 1 Z_15-symmetric variant: magma#0bcf3cca, |Aut| = 915 = 15*61. The order-15 subgroup of F_61* = (F_61*)^4 = {y : y^15 = 1} fixes the point 0 and acts by multiplication. In F_61 translation labeling, x*y = x + f(y - x) with f(y) = alpha(y)*y, where alpha takes ONLY the 4 Phi_10 roots, each on exactly one of the 4 (F_61*)^4-cosets. (See magma#0bcf3cca commentary for the explicit coset-to-root mapping.) * 1 Z_3-symmetric variant: magma#05fbff2c, |Aut| = 183 = 3*61. Stabilizer = order-3 subgroup {1, 13, 47}. In F_61 translation labeling, slope alpha is constant on the 20 Z_3-orbits of size 3, and takes 16 NON-Phi_10 slope values (avoiding {3, 27, 41, 52} entirely). The exact opposite of magma#0bcf3cca's Phi-only slope structure. * 30 translation-only variants: |Aut| = 61 (just Z_61). These split further by their *slope value set* in F_61 translation labeling: - The "Phi + inverse-paired tribe": 3 magmas (magma#e87f6bb9, magma#2426bbe0, magma#ec9ec6f6) use the same 12-element slope value set V = {3, 7, 10, 21, 27, 30, 32, 35, 41, 52, 55, 59} = (4 Phi_10 roots) union (4 non-Phi inverse-pairs {7,35}, {10,55}, {21,32}, {30,59} in F_61*). They differ by class-size distribution: e87f6bb9 has Phi-sum=44 (Phi-classes of size 11), 2426bbe0/ec9ec6f6 have Phi-sum=28 (Phi-classes of size 7). Note magma#0bcf3cca uses just the 4-element subset {3, 27, 41, 52} of V. - The other 27 magmas use larger slope-value sets, generally chaotic and unique per magma (|slope-values| ranges from 16 up to 38). A few pairs share slope-value sets (e.g. magma#2e8995b3 / magma#e1aef716, magma#b175c0f4 / magma#b8ecd854) - these are likely Galois-conjugate pairs. Slope-fingerprint distribution among the 30 translation-only magmas, by Phi-sum (= number of F_61* elements mapped to Phi_10-root slopes): Phi-sum=44 (1 magma), Phi-sum=28 (2), Phi-sum=6 (4), Phi-sum=4 (8), Phi-sum=2 (7), Phi-sum=0 (8 -- "Phi-free" translation-only variants, joining magma#05fbff2c structurally though without its Z_3 symmetry). Compare with size 41 (40 = 8*5 vs 60 = 4*15 has fewer divisors): size 41 has only 1 multi-stabilizer variant (Z_5, |Aut|=205) plus 7 translation-only Steiner-line magmas, and all 8 share a single 12-value slope set. Size 61 has both Z_15 and Z_3 multi-stabilizer variants and 26 distinct slope sets among 32 Steiner-line magmas, reflecting 60's richer divisor structure (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60). Smaller analogs at p ≡ 1 (mod 10): size 11 (just F_11 linear, 1 iso class). Larger: size 71 has |F_71*| = 70 = 2*5*7 with only Z_5 and Z_7 multi-stabilizer subgroups (no Z_15 analog). [text written by Claude]