Equation 677 Database

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Magma 2d5005e6916b…

magma 2d5005e6916b
Size
289
Isomorphism class hash
2d5005e6916ba6e431a8f0f36b35ada32b5b5bfe1115af31b207467bd704e353
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
no
Submitted by
dwrensha
Submitted at
2026-05-16 12:58:18
Display reorder
288,284,264,262,276,270,274,266,256,272,282,258,286,260,278,280,268,257,98,244,103,168,181,212,22,43,195,237,44,77,80,127,287,138,269,207,110,249,224,56,115,89,180,92,161,139,192,19,10,150,55,267,222,87,108,12,90,254,53,137,113,205,21,178,247,148,159,54,281,72,219,173,122,151,229,204,236,7,36,29,95,130,69,35,160,275,4,145,186,62,116,61,13,167,63,230,213,156,198,66,121,223,279,125,202,217,67,9,120,190,0,227,70,128,234,171,33,34,149,271,163,251,17,117,57,94,112,194,58,152,182,209,141,226,8,91,261,27,84,48,199,18,185,107,102,142,131,81,241,47,216,172,248,277,232,188,200,31,225,65,147,158,118,123,169,2,215,32,68,11,255,75,24,242,157,193,166,136,42,210,96,78,41,101,1,235,179,263,133,50,49,187,250,144,86,243,16,174,104,25,83,201,218,109,259,239,105,82,214,140,197,246,79,183,30,45,100,46,170,129,20,265,176,51,85,146,111,14,52,23,252,220,245,135,106,189,203,88,283,38,175,132,231,162,5,221,97,153,37,238,74,28,124,71,206,285,39,134,26,3,208,155,177,76,164,73,99,40,240,233,126,191,273,154,15,143,64,93,60,253,211,59,6,196,165,184,119,228,114 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 + 2α, b = 2 + 7α. Here β = 2 + 7α 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 12:59:10 · history