Equation 677 Database

Magma d51ca85e32d7…

magma d51ca85e32d7
Size
61
Isomorphism class hash
d51ca85e32d7d6406eaf3f1f85392c21d620d77306cf300360c9f340410aecb5
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
yes
Submitted by
bulk-import-memoryleak47
Submitted at
2026-04-23 21:19:30
Display reorder
0,1,47,14,59,60,9,43,7,57,10,37,12,45,40,48,4,29,50,2,26,27,32,58,8,25,49,31,35,15,28,41,54,19,22,13,20,42,30,24,39,44,36,23,5,18,3,55,53,38,56,11,52,46,51,17,6,34,16,33,21 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

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]

last edited by dwrensha at 2026-05-15 12:45:48 · history