Equation 677 Database

Size 65: 17 distinct Eq 677 magmas in the DB. Unlike sizes 41 and 61, 65 = 5*13 is composite. Crucially, the cyclotomic Phi_10(x) factors as (x+1)^4 mod 5 (single root with multiplicity 4) and is irreducible of degree 4 mod 13. By CRT, Phi_10 has NO roots in Z_65, so there is no "linear F_65 affine line" magma at this size. All 17 size-65 Eq 677 magmas are non-linear constructions. The 17 magmas split by idempotency: **9 fully idempotent magmas** -- Steiner-line over cyclic S(2, 5, 65). Each has 208 = 65*64/(5*4) size-5 sub-magmas (every block isomorphic to the F_5 affine line magma#e549b5f8). All 9 are translation-invariant over Z_65 = Z_5 x Z_13 (so |Aut| is a multiple of 65). They sub-classify by automorphism size and slope pattern: * 1 Z_4-symmetric variant: magma#eff27734, |Aut| = 260 = 4 * 65. Point-stabilizer is Z_4 = <34> = {1, 34, 51, 44} in Z_65*. In Z_65 translation labeling, slope alpha(y) = f(y)/y takes ONLY 4 values {3, 22, 35, 51}, each on a size-12 Z_4-coset. (Compare with magma#b7e8bf90 at size 41, magma#0bcf3cca at size 61: all are the "most symmetric Steiner-line" variants at their sizes.) * 8 translation-only variants: |Aut| = 65 (just Z_65). All 8 use the SAME 12-element slope value set on Z_65*: V = {3, 15, 21, 22, 31, 32, 34, 35, 44, 45, 51, 63}. Inside V: - 9 units form 4 inverse pairs {3,22}, {21,31}, {32,63}, {34,44} plus 1 self-inverse (51, since 51^2 = 1 mod 65) - 3 non-units {15, 35, 45} (all multiples of 5, hence non-invertible mod 65) The Z_4-symmetric magma#eff27734 uses the 4-element subset {3, 22, 35, 51} of V. The 8 translation-only magmas split by slope-class-size pattern: - Type A (5 magmas: magma#a09c4223, magma#86e94818, magma#4d60c34f, magma#bcf3ee89, magma#84076266): class sizes (6, 6, 6, 4, 4, 4, 4, 4, 4, 2, 2, 2) - sum 48 - Type B (2 magmas: magma#d8ba33b5, magma#2db4eae0): (8, 5, 5, 5, 5, 5, 5, 2, 2, 2, 2, 2) - Type C (1 magma: magma#62917c53): (9, 9, 9, 8, 2, 2, 2, 2, 2, 1, 1, 1) **8 non-idempotent magmas** -- F_5 x F_13 product-type constructions (already commented from a prior session): * magma#5e80e1c2dc82 (1 magma): pure direct product F_5 * F_13 of linear magmas * magma#b8dd5d8f5072, magma#b841b444f6b6, magma#f6a46900b44d, magma#22cb8afe1ff0, magma#3a1e3dea0200 (5 magmas): "F_5(2,4) base x F_13 fiber" bundles (5 fibers of 13) * magma#3e7f78778326 (1 magma): F_5 base x F_13 fiber with within-fiber smoothing * magma#17d8eb76b6bf (1 magma): contains a F_5(2,4) sub-magma + 5 fibers of 12 remaining elements Note the structural contrast: prime sizes (41, 61) admit linear F_p magmas AND Steiner-line magmas; composite sizes (65) admit ONLY Steiner-line idempotent magmas + non-idempotent fiber-bundle constructions. Compare with size 41 (|Aut|=205 multi-stab + 7 translation-only Steiner-line on a 12-element slope set), size 61 (|Aut|=915 Z_15-stab + |Aut|=183 Z_3-stab + 30 translation-only on 26 different slope sets). Size 65 has only 1 multi-stab (Z_4) and only 12 slope values across all 9 idempotent magmas -- but distributed across only 4 distinct slope-pattern types (eff27734 + Types A, B, C). The smaller variety at size 65 reflects 64 = phi(65) = 4*12 = order(Z_65*) being more constrained. Open question: what does the "Phi_10-roots-don't-exist-mod-N" constraint translate to in the algebraic structure of the magma? The 12-element slope set V suggests there's still some algebraic-number-theory residue, perhaps tied to Z_5-roots of Phi_2 = (x+1) lifting to mixed elements in Z_5 x Z_13. [text written by Claude]