Equation 677 Database

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Magma 2b8e3c0857ad…

magma 2b8e3c0857ad
Size
961
Isomorphism class hash
2b8e3c0857adde3bd7a822129cd9972fb3e4f89fdb5fe0b7d88d61100613a576
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
yes
Submitted by
b-reinke
Submitted at
2026-05-14 05:15:25
Display reorder
0,7,466,887,399,932,716,282,177,682,84,387,854,514,526,450,591,823,321,362,227,100,265,786,580,154,619,659,735,189,44,1,472,405,722,183,90,860,532,597,327,233,271,556,625,741,50,13,893,938,288,688,393,520,456,829,338,106,792,130,665,195,960,812,806,813,800,802,807,810,794,814,796,805,801,793,804,803,818,819,808,816,820,811,799,809,795,822,817,815,798,821,797,926,210,650,145,777,121,353,844,441,505,378,673,303,953,908,28,65,756,640,571,256,218,312,612,547,875,75,168,707,420,487,920,59,204,750,644,634,139,565,771,250,115,242,347,336,838,606,435,541,499,869,372,69,697,162,297,731,947,414,902,481,22,927,219,674,211,313,304,651,583,954,146,548,909,778,876,29,122,76,66,354,169,757,845,708,641,442,421,572,506,488,257,379,914,429,53,535,198,493,744,863,668,396,628,93,133,691,559,186,765,291,274,725,109,941,236,408,341,896,330,475,832,16,600,916,865,561,943,834,200,95,276,898,431,670,158,238,18,495,135,727,332,55,368,767,410,602,746,693,111,477,537,630,293,343,921,70,243,60,698,348,205,163,307,751,298,839,645,732,607,635,948,436,140,415,542,566,903,500,772,482,870,251,23,373,116,924,503,638,166,119,906,610,208,376,569,705,351,26,545,648,701,254,418,842,63,873,143,301,246,485,439,754,73,775,951,310,4,789,453,285,47,268,529,719,192,103,517,935,738,230,857,402,662,365,390,890,622,324,87,469,127,826,685,10,553,594,180,928,846,910,220,709,779,675,642,877,212,443,30,314,422,123,305,573,77,652,507,37,584,489,355,955,258,170,147,380,758,549,6,721,859,326,555,49,937,392,828,791,194,404,89,596,270,740,892,687,455,105,664,471,182,531,232,624,12,287,519,337,129,919,749,138,249,346,605,498,68,296,413,21,203,633,770,241,837,540,371,161,946,480,58,643,564,114,335,434,868,696,730,901,915,942,94,430,237,134,54,409,692,536,342,560,199,897,157,494,331,766,745,476,292,864,833,275,669,17,726,367,601,110,629,3,284,528,102,737,401,389,323,156,9,179,452,267,191,934,856,364,621,468,684,593,788,46,718,516,229,661,889,86,825,582,918,248,497,412,632,836,160,57,113,867,900,137,604,295,202,240,370,479,563,433,729,748,345,67,20,769,539,945,672,334,695,917,411,159,866,603,239,562,747,19,944,694,496,835,112,136,201,478,728,96,538,333,247,631,56,899,294,369,432,344,768,671,460,656,783,359,447,384,279,884,41,616,262,318,523,81,713,463,216,151,97,850,511,679,959,34,762,577,224,588,881,174,426,461,680,319,657,930,524,784,35,82,360,733,714,448,578,464,385,225,187,280,589,152,885,882,98,42,175,851,617,397,512,263,922,567,840,71,904,646,244,501,703,61,773,608,699,483,636,349,871,949,206,252,437,164,24,141,308,374,416,752,117,543,299,458,614,125,586,382,711,32,654,260,848,879,277,491,760,781,316,509,172,912,214,575,357,551,677,424,39,149,222,445,79,957,462,886,715,681,853,449,320,99,579,658,43,465,931,176,386,525,852,226,785,618,188,36,398,281,83,513,590,361,264,153,734,925,144,352,504,302,27,639,217,546,167,486,649,120,440,702,907,755,255,611,74,419,209,776,843,377,952,64,570,311,874,706,913,534,743,395,132,185,273,940,340,474,599,52,522,667,92,558,290,108,407,329,15,428,197,862,627,690,764,724,235,895,831,923,165,609,568,25,700,841,142,484,72,309,637,905,375,350,647,417,872,245,753,950,502,118,207,704,544,253,62,300,438,774,5,325,936,790,88,739,454,470,231,286,128,858,48,827,403,269,686,663,530,11,366,720,554,391,193,595,891,104,181,623,518,2,101,388,8,266,855,467,787,515,888,581,527,400,155,451,933,620,592,717,660,824,283,736,322,178,190,363,683,45,228,85,459,358,278,615,552,492,126,678,761,587,425,782,383,40,317,712,150,510,33,223,173,655,446,883,261,80,215,849,958,576,880,457,585,31,847,490,315,911,356,423,221,956,124,710,259,306,780,171,574,676,148,78,613,381,653,878,759,508,213,550,38,444,929,394,272,473,521,557,406,427,626,723,830,742,184,339,51,91,107,14,861,763,894,533,131,939,598,666,289,328,196,689,234 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 = 15. Each line is isomorphic to magma#7ba82cf6 (= x ◇ y = 17x + 15y mod 31, the F_31 linear magma with α = 15 ∈ {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=15) × F_31(α=15), a single F_31²-linear quasigroup with (x₁, x₂) ◇ (y₁, y₂) = ((1 − 15)·x_i + 15·y_i mod 31)_(i=1,2) Sibling magmas at size 961, all 4 constant-α direct products: magma#fc322ba7, magma#5f44e78d, magma#ba90950a, plus this one (α=15). 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