Equation 677 Database

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Magma a586ea985273…

magma a586ea985273
Size
289
Isomorphism class hash
a586ea9852731c2e66d0fd9071e113176fb0ba0da12b9938f45faa204e7e3901
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
no
Submitted by
dwrensha
Submitted at
2026-05-16 12:58:16
Display reorder
288,284,179,49,74,268,227,17,243,106,163,90,122,211,195,33,138,25,159,139,183,109,60,213,275,7,11,277,72,63,249,191,283,12,203,189,54,172,34,250,78,241,278,144,238,77,108,178,170,199,185,82,261,111,216,40,14,196,64,239,240,59,48,157,42,69,117,120,98,30,75,133,232,255,58,286,212,73,145,65,256,85,136,80,182,9,259,166,84,56,143,274,267,44,47,123,269,279,197,224,167,252,251,135,156,237,6,226,102,52,287,220,192,247,91,3,233,15,134,41,76,10,16,208,23,265,175,229,88,37,125,27,207,164,8,79,276,242,217,31,253,234,28,151,127,263,150,22,236,258,181,68,107,130,244,214,35,112,99,168,4,126,264,5,165,97,116,177,71,105,187,1,154,18,61,173,262,174,225,92,38,128,169,62,162,147,222,114,271,161,96,149,228,110,55,83,81,198,248,89,152,100,19,272,155,115,21,87,51,142,132,260,184,190,121,137,280,193,230,29,113,235,36,0,273,231,119,202,282,210,66,140,204,118,86,270,221,176,219,266,254,215,188,20,194,205,94,93,201,50,160,186,70,257,124,57,39,53,24,32,245,223,101,281,141,95,2,104,180,26,200,43,171,158,146,45,103,209,218,131,148,153,129,67,206,246,46,285,13 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

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]

last edited by dwrensha at 2026-05-16 12:59:09 · history