Equation 677 Database

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Magma ba90950a375d…

magma ba90950a375d
Size
961
Isomorphism class hash
ba90950a375d5ba801c3af4f51afc3935a0f0d9c653ccb0c888cc9d3bd90d3da
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
yes
Submitted by
dwrensha
Submitted at
2026-05-13 12:18:05
Display reorder
0,3,414,30,701,51,485,732,502,423,790,858,159,212,359,799,537,899,108,765,898,442,757,275,521,381,235,389,818,131,932,1,422,107,388,50,358,274,17,789,764,817,484,798,520,413,857,897,130,731,536,380,29,158,441,931,501,913,234,700,211,756,924,827,464,254,747,249,333,320,138,307,347,715,591,606,170,664,678,576,955,621,198,549,65,651,636,185,457,847,43,565,83,553,110,53,277,792,820,801,416,885,719,383,161,934,901,703,759,425,391,361,5,767,487,508,860,118,524,32,444,504,237,214,96,56,780,804,888,386,937,691,428,364,770,511,121,20,507,217,113,280,823,404,722,149,904,762,394,8,490,863,527,447,225,960,881,878,284,875,876,296,880,872,882,873,295,293,289,877,294,869,287,879,883,870,283,292,297,290,871,286,285,874,288,291,916,35,192,72,628,685,613,354,327,739,834,557,449,658,205,583,177,707,145,241,471,90,839,643,541,947,671,598,299,340,261,928,174,47,704,189,142,69,238,625,468,682,87,610,851,351,640,324,538,736,959,831,668,554,595,461,311,655,337,202,258,580,2,778,730,233,372,412,440,421,816,535,699,273,856,930,106,483,379,210,16,896,500,402,797,28,755,788,129,912,49,519,157,925,321,171,550,44,748,716,956,186,828,139,665,66,566,250,592,622,458,465,308,679,652,84,334,607,199,848,255,348,577,637,926,957,251,653,349,45,666,466,200,322,187,593,85,578,734,67,309,849,172,829,623,335,638,717,567,680,256,551,140,459,608,101,376,775,222,727,13,230,893,369,497,409,399,437,794,418,25,813,752,532,785,696,126,270,909,868,61,942,516,103,154,480,915,738,144,597,684,204,642,34,833,240,298,612,582,540,191,556,470,339,353,176,946,71,448,89,260,326,706,670,627,657,853,922,619,318,455,168,477,547,305,41,676,745,649,713,81,953,331,183,604,825,196,136,845,662,267,78,345,563,574,247,634,589,917,355,206,91,300,614,558,146,948,36,313,569,840,341,686,450,242,672,178,740,163,629,262,599,644,472,584,73,835,708,542,923,306,954,846,248,169,650,826,346,620,42,332,663,635,463,714,197,564,319,677,184,253,590,548,82,137,575,456,746,605,64,100,496,531,515,12,432,908,375,408,784,117,229,24,867,774,398,695,153,892,812,60,221,436,125,479,368,751,941,726,808,269,98,810,373,749,772,529,219,782,724,693,10,123,227,282,890,906,366,865,494,58,406,939,396,513,434,115,806,151,430,492,22,94,417,689,488,505,793,162,362,525,111,886,760,509,223,821,935,6,18,54,720,426,861,215,802,902,768,445,278,384,392,119,927,467,735,336,141,350,594,173,681,958,201,68,639,460,46,86,830,257,252,323,310,718,609,667,579,624,552,654,188,850,568,99,783,891,940,431,773,124,495,116,811,725,907,397,478,530,228,59,807,374,694,367,514,23,220,268,407,152,750,11,866,435,919,180,616,601,315,837,452,193,165,133,474,842,544,659,302,264,38,75,688,357,742,560,646,571,710,244,93,631,950,586,328,929,733,360,443,819,702,859,109,382,4,503,800,758,132,52,160,766,236,415,424,523,276,933,486,213,884,390,31,791,900,522,914,239,352,669,203,190,88,737,312,175,626,641,555,259,143,611,945,656,48,469,325,596,581,70,852,832,338,705,683,539,462,918,147,687,630,836,301,570,179,473,356,949,451,263,709,615,841,741,585,207,37,243,600,543,559,342,164,74,92,314,673,645,920,194,303,561,951,316,843,674,245,181,166,265,647,587,838,545,343,79,617,134,39,572,329,453,660,743,632,602,475,76,711,33,534,15,911,411,105,27,777,698,895,63,439,482,754,729,272,499,518,420,378,787,232,855,401,156,815,209,128,371,944,796,97,692,493,150,528,889,512,809,9,57,429,218,905,433,387,122,405,491,781,365,114,763,226,938,21,723,864,805,771,281,395,102,894,419,127,104,728,400,533,62,377,370,26,271,155,14,438,786,943,776,498,814,910,481,231,795,697,517,208,410,753,854,95,363,822,862,385,506,761,55,769,403,526,936,216,393,779,510,721,446,690,112,7,803,120,148,224,427,279,489,887,19,903,921,675,824,573,476,952,266,618,744,195,246,546,330,77,317,648,135,633,304,182,344,454,712,844,588,40,603,562,167,80,661 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 = 27. Each line is isomorphic to magma#543c5479 (= x ◇ y = 5x + 27y mod 31, the F_31 linear magma with α = 27 ∈ {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=27) × F_31(α=27), a single F_31²-linear quasigroup with (x₁, x₂) ◇ (y₁, y₂) = ((1 − 27)·x_i + 27·y_i mod 31)_(i=1,2) Sibling magmas at size 961, all 4 constant-α direct products: magma#fc322ba7, magma#2b8e3c08, magma#5f44e78d, plus this one (α=27). 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