Linear magma over the extension field F_289 = F_17[α]/⟨α² − 3⟩.
Operation: x ◇ y = a·x + b·y in F_289 with a = 8 + 15α, b = 6 + 5α. Here β = 6 + 5α is a root of Φ_2_5(x) = x⁴ + x³ + 2x² + 2x + 1 in F_289, and α_coef = -β³ - β - 1. This is Pace Nielsen's "Type 2" non-fully-idempotent linear 677 magma family.
Note: 10 ∤ (17² − 1) = 288, so F_289 has no primitive 10th roots of unity → no Type-1 (translation-invariant fully-idempotent) linear 677 magma exists at this size. But Φ_2_5 has 4 roots in F_289 (forming 2 Galois-conjugate orbits under Frobenius), giving 2 distinct iso classes of Type-2 magmas at this size.
F_289 is the proper degree-2 extension of F_17; do NOT confuse with the ring Z/289Z (which has zero divisors and is NOT a field).
Size 289, not fully idempotent, right-cancellative. This iso class is one of the two known Eq 677 magmas at size 289 in the DB.
[text written by Claude]
dwrensha · 2026-05-16 12:59:09