Equation 677 Database

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Magma fc322ba743f6…

magma fc322ba743f6
Size
961
Isomorphism class hash
fc322ba743f69d20d52473266bda3e6e3c600bf85e9ecf01f012a16370d71310
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
yes
Submitted by
b-reinke
Submitted at
2026-05-14 05:25:54
Display reorder
0,5,761,499,276,778,48,870,523,800,664,429,142,216,941,571,97,330,621,477,454,687,179,255,205,402,606,902,727,826,75,1,329,498,453,47,254,799,605,141,825,570,34,620,275,686,869,204,663,901,215,74,96,760,476,777,178,552,401,428,726,940,925,516,35,817,129,558,638,674,192,889,92,748,765,540,416,958,317,441,272,593,843,22,293,857,651,233,114,464,166,389,714,960,345,357,358,339,335,340,349,351,341,347,344,352,338,361,348,363,346,353,337,359,362,356,336,364,355,350,354,343,342,360,3,729,374,181,479,99,218,666,872,278,7,828,608,257,456,332,943,431,525,780,733,77,904,207,689,623,573,144,802,50,501,922,955,711,413,386,537,163,792,491,745,111,89,230,886,648,189,854,701,290,635,19,555,840,126,590,814,269,62,438,513,314,923,90,841,712,887,591,387,190,270,164,702,439,492,636,315,112,556,956,231,127,414,649,815,538,855,63,763,291,514,746,20,370,866,937,683,723,302,425,617,398,31,549,567,175,852,774,138,473,602,757,796,123,251,71,44,242,450,898,495,660,326,201,366,613,770,247,656,719,563,753,446,862,394,134,67,322,298,171,822,894,933,27,469,40,197,421,848,119,521,679,545,598,238,371,568,72,938,823,243,724,139,899,426,603,661,399,797,202,550,252,867,176,45,684,775,451,303,474,496,618,758,327,32,124,914,282,529,148,103,460,211,703,11,784,806,222,306,693,378,832,737,54,670,947,627,155,612,81,505,876,405,577,483,261,908,916,485,878,584,949,739,695,808,705,105,284,263,407,83,629,56,380,224,13,462,531,910,579,507,157,672,834,308,786,213,150,372,662,452,73,798,475,824,551,619,725,868,328,900,46,95,604,776,569,400,304,939,203,497,244,253,759,140,177,33,427,685,929,804,625,906,782,945,259,9,668,481,731,52,575,209,735,403,458,830,874,101,376,503,146,691,79,527,334,610,280,220,183,2,430,180,762,217,206,277,572,607,49,331,728,524,478,76,665,688,6,143,256,500,942,373,779,98,903,871,622,827,801,455,917,264,580,879,84,158,950,57,835,696,225,787,706,433,151,285,911,486,408,508,585,630,643,740,381,309,809,14,214,106,532,369,601,682,122,424,43,30,897,174,325,137,865,756,722,250,616,241,548,494,851,200,472,936,795,301,70,397,449,566,659,773,926,749,294,36,541,652,130,959,115,639,442,167,193,594,715,93,23,517,766,858,818,417,234,559,318,465,675,273,390,890,844,913,53,147,626,210,80,783,404,305,260,831,281,669,102,184,732,504,805,576,692,907,736,528,946,459,611,10,875,221,482,377,928,392,467,236,860,25,596,444,931,543,751,892,677,561,820,519,717,169,117,654,296,846,245,320,419,768,65,195,641,132,38,4,829,905,375,258,690,480,333,574,219,432,803,873,781,502,8,78,730,609,208,182,457,624,100,944,145,667,526,51,279,734,918,788,644,581,434,382,85,286,810,951,487,185,836,509,533,226,631,265,707,741,880,152,310,159,912,15,58,409,107,697,586,368,547,121,199,29,794,324,396,755,658,615,600,493,423,471,896,300,136,448,721,772,240,681,850,42,935,173,69,864,565,249,924,440,816,842,637,856,888,113,764,388,957,515,271,128,21,673,650,91,463,539,713,316,64,592,557,292,191,232,747,165,415,927,168,235,295,595,319,542,94,676,131,518,391,116,859,845,443,418,750,194,560,37,716,466,653,24,274,930,767,891,640,819,921,700,412,18,162,125,744,268,229,512,188,954,289,385,554,791,589,110,61,885,313,853,710,634,536,839,490,813,88,437,647,367,135,198,771,323,849,657,172,522,564,895,546,447,28,239,395,41,614,68,422,248,299,120,720,793,680,754,934,599,863,470,365,893,246,468,562,420,861,520,66,597,170,642,932,655,39,752,847,393,678,321,237,821,769,26,718,196,445,118,133,544,297,920,109,17,312,743,633,511,489,288,436,790,699,60,161,882,267,535,187,812,384,646,588,411,884,154,709,228,838,953,87,553,915,223,583,530,694,506,104,833,406,212,55,484,12,948,909,807,156,283,307,82,149,379,877,461,738,578,704,671,262,785,628,919,186,311,645,510,883,435,227,59,86,266,108,811,742,587,488,153,789,837,160,582,534,16,383,632,410,287,708,698,952,881 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

Size-961 = 31² idempotent right-cancellative magma satisfying Eq 677 and Eq 255, in the AG(2, 31) line family. The 992 = 32 × 31 size-31 sub-magmas are exactly the lines of AG(2, 31); every pair of distinct points lies on a unique line. Construction parameter: ALL 32 parallel classes use the SAME α = 29. Each line is isomorphic to magma#7ee5b4b8 (= x ◇ y = 3x + 29y mod 31, the F_31 linear magma with α = 29 ∈ {15, 23, 27, 29} = primitive 10th roots of unity in F_31 = roots of Φ_10 mod 31). Equivalently, this is the medial direct product F_31(α=29) × F_31(α=29), a single F_31²-linear quasigroup with (x₁, x₂) ◇ (y₁, y₂) = ((1 − 29)·x_i + 29·y_i mod 31)_(i=1,2) Sibling magmas at size 961, all 4 constant-α direct products: magma#2b8e3c08, magma#5f44e78d, magma#ba90950a, plus this one (α=29). Display reorder coordinates each point as F_31² (a, b) at index 31·a + b. F_31 addition is computable in the magma itself via the identity x + y = T(R_0⁻¹(x), L_0⁻¹(y)) where R_0(x) = T[x, 0] (= ×(1 − α) in F_31) and L_0(x) = T[0, x] (= ×α). This lets us recover canonical → F_31² labels by BFS over the ⟨R_0, L_0⟩ action on canonical 1 (giving the horizontal axis F_31 × {0}), then on a chosen vertical generator (giving the vertical axis), then combining via the addition formula above. Under this reorder the Cayley table is fully (Z/31)²-translation-invariant AND every cell matches the medial-product formula exactly — visually a clean rainbow-diagonal pattern. [text written by Claude]

last edited by dwrensha at 2026-05-14 23:41:03 · history