Translation-only Steiner-line magma at size 65 (Type A slope pattern - 5 magmas).
Size 65 = 5 * 13, fully idempotent, RC. 208 size-5 sub-magmas on a cyclic S(2, 5, 65) design. Each block = the F_5 affine line magma#e549b5f8. |Aut(M)| = 65 (only Z_65 translation, no multiplicative stabilizer).
In the suggested reorder, Z_65 labeling makes translation x -> x+1 an automorphism. Operation: x*y = x + f(y - x) mod 65.
Slope analysis on Z_65* (48 units): alpha takes 12 distinct values {3, 15, 21, 22, 31, 32, 34, 35, 44, 45, 51, 63}, with the following class sizes (Type A pattern):
3 slopes on size-6 classes (sum 18)
6 slopes on size-4 classes (sum 24)
3 slopes on size-2 classes (sum 6)
Total: 18 + 24 + 6 = 48 = |Z_65*|.
Three of the 12 slopes ({15, 35, 45}) are non-units mod 65 (multiples of 5); the other 9 are units and form 4 inverse pairs in Z_65* {3,22}, {21,31}, {32,63}, {34,44} plus the self-inverse 51 (= -14, 51^2 = 1 mod 65).
This 12-value slope set is exactly the one obtained by taking all Z_65* elements that "could be Eq 677 multipliers" plus the 3 multiples of 5. Compare with magma#eff27734 (Z_4-symmetric) which uses only the 4-element subset {3, 22, 35, 51} as slopes - that subset is the Z_4 = <34> stabilizer multiplied by representative slope values.
This (Type A) slope-class-size pattern is shared by 5 magmas at size 65: this one and magma#86e94818, magma#4d60c34f, magma#bcf3ee89, magma#84076266. They differ in how the 12 slope values are assigned to specific Z_65*-subsets. Other Type B/C patterns (also using the same 12-value slope set) are: Type B (2 magmas: magma#d8ba33b5, magma#2db4eae0), Type C (1 magma: magma#62917c53).
[text written by Claude]
dwrensha · 2026-05-15 13:00:31
Steiner-line magma on a cyclic S(2, 5, 65) design, translation-invariant over Z_65.
Size 65 = 5 * 13, fully idempotent, RC. 208 = 65*64/(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 16 blocks. |Aut(M)| = 65 -- only the Z_65 = Z_5 x Z_13 translation subgroup; no multiplicative stabilizer.
Note: unlike sizes 41 and 61 (primes ≡ 1 mod 10 with linear magmas), 65 is composite (5 * 13) and Phi_10 has NO roots in Z_65: the cyclotomic Phi_10(x) factors as (x+1)^4 mod 5 and is irreducible of degree 4 mod 13, so by CRT there are no simultaneous roots mod 65. Consequently, there are no "linear F_65 affine line" magmas at this size. Every size-65 idempotent Eq 677 magma is a (non-linear) Steiner-line magma.
In the suggested reorder, Z_65 labeling makes translation x -> x+1 an automorphism. The operation is x*y = x + f(y - x) mod 65 with f a non-linear permutation of Z_65 satisfying f(0) = 0.
Structural notes:
- The Z_5 additive subgroup {0, 13, 26, 39, 52} of Z_65 forms a size-5 SUB-MAGMA isomorphic to magma#e549b5f8 (this is automatic: f preserves multiples-of-13 in Z_65). This is one of the 208 Steiner blocks, made "canonical" by translation invariance.
- The Z_13 additive subgroup {0, 5, 10, ..., 60} of Z_65 does NOT form a sub-magma: f does not preserve multiples of 5.
Slope analysis on Z_65* (the 48 units mod 65): alpha(y) = f(y) * y^{-1} mod 65 takes 12 distinct values:
alpha in {3, 15, 21, 22, 31, 32, 34, 35, 44, 45, 51, 63}
Class sizes: (6, 6, 6, 4, 4, 4, 4, 4, 4, 2, 2, 2) totaling 48 = |Z_65*|.
Three of the 12 slopes (alpha in {15, 35, 45}) are NOT units mod 65 (each is a multiple of 5); the other 9 are units. So f maps some Z_65*-elements to multiples of 5 in Z_65 \ Z_65*. This is unlike the prime-size case where slopes are always units.
This magma is one of (at least) 9 fully idempotent Steiner-line magmas at size 65 in the DB. The size-65 family also includes 8 non-idempotent variants (F_5 x F_13 direct products or F_5 base x F_13 fiber bundles).
[text written by Claude]
dwrensha · 2026-05-15 13:03:50
dwrensha · 2026-05-15 13:00:31