Equation 677 Database

Magma 18304af5f7ca…

magma 18304af5f7ca
Size
77
Isomorphism class hash
18304af5f7caa048a8c850a4737239c8b810350baddda3284c82a3ed696abf70
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
no
Submitted by
dwrensha
Submitted at
2026-06-10 13:28:22
Display reorder
76,62,65,61,64,63,60,75,37,17,27,57,47,6,68,39,19,29,59,49,8,69,34,14,24,55,44,4,71,35,15,25,56,45,5,66,33,13,23,54,43,3,72,32,12,22,52,42,2,67,38,18,28,58,48,7,74,31,11,21,51,41,1,73,36,16,26,53,46,9,70,30,10,20,50,40,0 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

Size-77 twisted fiber-bundle magma satisfying Eq 677 (and Eq 255). Carrier Z/11 × F_7 with operation (x, s) ◇ (y, t) = ( x + 8·(y − x) mod 11 , α_d·s + β_d·t in F_7 ), where d = y − x mod 11. The base coefficient c = 8 is a primitive 10th root of unity mod 11 (Φ₁₀(8) = 8⁴ − 8³ + 8² − 8 + 1 ≡ 0 mod 11), which is exactly what Eq 677 forces on the base operation x + c·(y − x). The fiber field is F_7 = Z/7Z (prime field). The fiber coefficient pair (α_d, β_d) depends on the base difference d: d = 0: (α, β) = (4, 3) d = 1: (α, β) = (4, 1) d = 2: (α, β) = (4, 1) d = 3: (α, β) = (4, 1) d = 4: (α, β) = (4, 1) d = 5: (α, β) = (4, 1) d = 6: (α, β) = (4, 1) d = 7: (α, β) = (4, 1) d = 8: (α, β) = (4, 1) d = 9: (α, β) = (4, 1) d = 10: (α, β) = (4, 1) Eq 677 holds iff the pairs satisfy, for every d (writing e₁ = −d, e₂ = cd, e₃ = d(1 − c + c²), e₄ = −d/c, all mod 11): β_e₄·α_e₃ + β_e₄·β_e₃·α_e₂·β_e₁ = 1 and α_e₄ + β_e₄·β_e₃·(α_e₂·α_e₁ + β_e₂) = 0, with the d = 0 pair therefore satisfying the standard linear-677 conditions α₀β₀(1 + β₀²) = 1, α₀ + α₀²β₀² + β₀³ = 0. This family strictly generalizes the quadratic-residue-class bundles searched earlier (where (α_d, β_d) was constant on QR/non-QR classes): here the pair may vary with d arbitrarily. Found June 2026 by an exhaustive constraint solve over all per-d coefficient assignments; Eq 677 verified directly on the full Cayley table. The magma is right-cancellative; not idempotent; it satisfies Eq 255 (consistent with the 677 ⇒ 255 finite conjecture). [text written by Claude]

last edited by dwrensha at 2026-06-10 13:28:23 · history