Equation 677 Database

← all  ·  ← by size  ·  ← size 961

Magma 5f44e78d513f…

magma 5f44e78d513f
Size
961
Isomorphism class hash
5f44e78d513f3d81773b40bb356c073357ccb23ac7f2f7fdab865d1855e6aaeb
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
yes
Submitted by
b-reinke
Submitted at
2026-05-14 05:18:41
Display reorder
0,2,5,17,14,11,8,26,23,20,461,458,455,452,449,446,443,440,437,434,431,428,425,422,419,416,413,410,407,404,401,1,417,435,453,27,3,414,432,450,24,6,411,429,447,21,18,408,426,444,462,15,405,423,441,459,12,402,420,438,456,9,775,760,745,823,808,793,778,763,748,826,811,796,781,766,751,829,814,799,784,769,754,832,817,802,787,772,757,835,820,805,790,204,129,147,165,183,201,126,144,162,180,198,123,141,159,177,195,213,138,156,174,192,210,135,153,171,189,207,132,150,168,186,339,357,375,393,318,336,354,372,390,315,333,351,369,387,312,330,348,366,384,309,327,345,363,381,399,324,342,360,378,396,321,663,681,699,717,735,660,678,696,714,732,657,675,693,711,729,654,672,690,708,726,651,669,687,705,723,741,666,684,702,720,738,929,13,463,448,433,418,403,16,22,451,436,421,406,19,25,454,439,424,409,7,28,457,442,427,412,4,10,460,445,430,415,662,659,656,653,650,740,737,734,731,728,725,722,719,716,713,710,707,704,701,698,695,692,689,686,683,680,677,674,671,668,665,228,246,264,282,300,225,243,261,279,297,222,240,258,276,294,219,237,255,273,291,216,234,252,270,288,306,231,249,267,285,303,927,852,870,888,906,924,849,867,885,903,921,846,864,882,900,918,843,861,879,897,915,840,858,876,894,912,837,855,873,891,909,229,307,292,277,262,247,232,217,295,280,265,250,235,220,298,283,268,253,238,223,301,286,271,256,241,226,304,289,274,259,244,773,770,767,764,761,758,755,752,749,746,743,833,830,827,824,821,818,815,812,809,806,803,800,797,794,791,788,785,782,779,776,638,635,632,629,626,623,620,617,614,611,608,605,602,599,596,593,590,587,584,581,578,575,572,569,566,563,560,557,647,644,641,338,335,332,329,326,323,320,317,314,311,308,398,395,392,389,386,383,380,377,374,371,368,365,362,359,356,353,350,347,344,341,926,923,920,917,914,911,908,905,902,899,896,893,890,887,884,881,878,875,872,869,866,863,860,857,854,851,848,845,842,839,836,928,913,898,883,868,853,838,916,901,886,871,856,841,919,904,889,874,859,844,922,907,892,877,862,847,925,910,895,880,865,850,205,190,175,160,145,130,208,193,178,163,148,133,211,196,181,166,151,136,214,199,184,169,154,139,124,202,187,172,157,142,127,48,66,84,102,120,45,63,81,99,117,42,60,78,96,114,39,57,75,93,111,36,54,72,90,108,33,51,69,87,105,30,484,469,547,532,517,502,487,472,550,535,520,505,490,475,553,538,523,508,493,478,556,541,526,511,496,481,466,544,529,514,499,203,200,197,194,191,188,185,182,179,176,173,170,167,164,161,158,155,152,149,146,143,140,137,134,131,128,125,122,212,209,206,227,224,221,218,215,305,302,299,296,293,290,287,284,281,278,275,272,269,266,263,260,257,254,251,248,245,242,239,236,233,230,340,325,310,388,373,358,343,328,313,391,376,361,346,331,316,394,379,364,349,334,319,397,382,367,352,337,322,400,385,370,355,483,501,519,537,555,480,498,516,534,552,477,495,513,531,549,474,492,510,528,546,471,489,507,525,543,468,486,504,522,540,465,960,951,946,935,957,952,944,959,931,932,947,940,937,934,930,933,950,955,948,945,958,936,954,956,941,953,943,939,949,938,942,49,34,112,97,82,67,52,37,115,100,85,70,55,40,118,103,88,73,58,43,121,106,91,76,61,46,31,109,94,79,64,774,792,810,828,753,771,789,807,825,750,768,786,804,822,747,765,783,801,819,744,762,780,798,816,834,759,777,795,813,831,756,664,742,727,712,697,682,667,652,730,715,700,685,670,655,733,718,703,688,673,658,736,721,706,691,676,661,739,724,709,694,679,640,625,610,595,580,565,643,628,613,598,583,568,646,631,616,601,586,571,649,634,619,604,589,574,559,637,622,607,592,577,562,47,44,41,38,35,32,29,119,116,113,110,107,104,101,98,95,92,89,86,83,80,77,74,71,68,65,62,59,56,53,50,482,479,476,473,470,467,464,554,551,548,545,542,539,536,533,530,527,524,521,518,515,512,509,506,503,500,497,494,491,488,485,639,564,582,600,618,636,561,579,597,615,633,558,576,594,612,630,648,573,591,609,627,645,570,588,606,624,642,567,585,603,621 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 = 23. Each line is isomorphic to magma#fdf14b96 (= x ◇ y = 9x + 23y mod 31, the F_31 linear magma with α = 23 ∈ {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=23) × F_31(α=23), a single F_31²-linear quasigroup with (x₁, x₂) ◇ (y₁, y₂) = ((1 − 23)·x_i + 23·y_i mod 31)_(i=1,2) Sibling magmas at size 961, all 4 constant-α direct products: magma#fc322ba7, magma#2b8e3c08, magma#ba90950a, plus this one (α=23). 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