Equation 677 Database

← all  ·  ← by size  ·  ← size 721

Magma 72c593f7f673…

magma 72c593f7f673
Size
721
Isomorphism class hash
72c593f7f6731f7550580a10f30d1ba3fa20ca951b6f19977bb7e72de7f54f7f
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
no
Fiber matrix
symmetric: yes · normal: yes · rank: 1 (nullity 720) what is this?
Submitted by
dwrensha
Submitted at
2026-09-27 18:48:40
Display reorder
608,661,280,187,500,595,606,120,672,575,567,574,431,281,308,618,305,603,122,137,594,223,676,483,469,514,373,275,420,646,311,363,182,337,37,199,702,495,284,249,6,148,590,664,15,457,434,601,299,530,663,111,115,532,577,209,265,686,411,45,255,367,570,548,654,521,549,562,385,196,578,694,569,715,458,161,348,426,615,56,157,410,276,318,259,630,123,303,92,113,576,64,644,0,609,243,240,300,54,648,545,206,598,246,360,480,675,117,433,95,503,510,132,623,395,69,98,589,190,505,626,30,218,93,504,534,283,690,335,109,194,301,478,68,707,36,175,464,27,57,295,647,509,579,310,228,378,368,688,153,439,116,270,372,554,680,353,415,128,38,245,211,660,515,103,87,288,306,180,621,383,121,358,216,293,126,720,177,242,267,307,172,332,628,129,357,237,63,154,566,699,262,296,81,179,466,302,614,605,321,219,167,490,102,650,317,327,279,361,528,560,685,244,29,90,413,124,9,653,10,427,352,143,160,386,652,232,585,89,12,269,59,679,46,139,538,149,11,344,673,183,561,452,22,484,168,625,429,309,297,547,257,596,627,25,351,291,294,558,44,697,35,525,164,8,354,174,678,208,369,225,461,106,28,635,477,399,110,407,717,518,640,527,145,334,511,600,572,706,599,519,316,204,611,462,649,214,260,526,583,166,162,713,581,248,550,485,406,73,701,539,97,88,198,233,444,634,250,3,388,131,312,362,684,471,224,322,565,418,432,613,274,543,188,173,552,356,670,359,230,482,523,516,524,632,171,555,544,449,136,229,698,557,487,176,234,239,380,705,551,375,85,191,215,392,617,51,573,273,455,546,408,708,587,65,370,119,564,277,712,20,34,714,258,442,186,669,453,451,94,210,47,156,616,435,387,422,343,496,414,643,347,421,328,443,76,138,641,371,14,568,155,52,96,677,268,381,152,325,384,450,666,405,212,231,541,227,271,639,441,403,580,313,142,474,620,41,254,84,535,582,49,638,99,463,91,264,118,512,622,365,55,400,67,66,114,693,61,315,158,559,42,108,709,298,272,134,571,184,326,624,533,60,83,391,16,144,658,489,200,86,48,5,374,665,716,339,604,467,588,253,687,238,393,476,437,178,235,637,141,181,556,397,460,39,683,447,79,440,419,448,18,662,389,236,446,479,540,468,710,70,163,201,331,396,205,651,593,481,146,43,424,241,636,401,151,304,355,100,350,642,292,475,80,197,430,542,689,256,345,382,252,112,13,656,417,591,428,497,287,338,700,75,499,470,491,21,498,645,4,409,346,607,130,192,657,286,127,488,425,342,289,633,202,537,416,553,390,719,703,323,40,507,58,251,438,711,195,74,340,72,31,584,674,377,133,520,185,221,247,655,159,2,610,107,366,33,631,165,50,82,349,26,314,671,563,169,213,32,412,150,692,508,531,586,77,203,1,668,423,193,364,529,454,492,691,220,278,592,282,263,456,704,501,333,376,101,330,217,612,147,24,104,517,71,320,695,189,19,494,379,472,398,696,465,445,394,473,336,486,681,105,506,261,17,502,402,619,135,493,207,718,62,78,659,226,513,170,319,7,536,629,459,597,404,222,522,23,682,341,266,140,53,324,602,667,329,290,285,125,436 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

Monogenic cyclic-affine Eq677 magma of order 721. On Z/721, in construction labels, x*y = 571x + 24y + 1 (mod 721). Element 0 generates the whole magma; there are 612 generators and 1 idempotent element(s). For any element r, the generated submagma has size 721/gcd(721, (571+24-1)r+1). Both coefficients are units modulo 721, so this is a quasigroup (left- and right-cancellative); its fiber matrix is all ones. The affine Eq677 conditions are ab(1+b^2)=1, a+a^2*b^2+b^3=0, and (b^2(a+1)+b+1)c=0 modulo n. These hold for the displayed parameters. Independently checked all 519841 Eq677 instances, all 721 Eq255 instances, and full set closure from generator 0 directly in the saved table. This is not a counterexample to Eq677 => Eq255. Found while enumerating monogenic cyclic-affine models of orders below 1000, working upward in order. These are standard affine constructions, added to fill gaps in the database. Construction labels may differ from the server canonical labels. Computed with OpenAI Codex working with dwrensha, 2026-09-27.

last edited by dwrensha at 2026-09-27 18:48:40 · history