F_29² Type-1 affine line magma at size 841.
Size 841 = 29², fully idempotent, RC. Linear over the extension field F_29² = F_29[α]/⟨α² − 2⟩.
Operation: x ◇ y = (1−β)·x + β·y in F_29², with β a primitive 10th root of unity in F_29² (= root of Phi_10(x) = x⁴ − x³ + x² − x + 1). F_29² has |F_29²*| = 840 = 2³·3·5·7, divisible by 10, so 4 primitive 10th roots exist; up to Galois conjugacy these form 2 orbits of 2, giving 2 distinct iso classes at size 841 (this magma corresponds to one of the two Galois orbits; the other is magma#cc0cd58750be).
This is the "AG(2, 29) affine line over F_29²" magma. Sub-magma structure: NO size-29 sub-magmas (since the Phi_10 root β lies in F_29² \ F_29, the F_29 prime subfield is not preserved by the operation). Every pair of distinct elements generates the full 841-element magma.
Companion to the other F_29² linear magmas at size 841 in the DB: magma#cc0cd58750be (the other Type-1 iso class) and the two Type-2 magmas magma#8469938fe79d, magma#630bb9976070 (Phi_2_5-root constructions).
The size-841 family also includes magma#ad16fb07 (an exceptional non-standard construction, possibly the Zassenhaus near-field of order 841) and various other entries — see size-841 page commentary.
[text written by Claude]
dwrensha · 2026-05-20 19:28:51