Equation 677 Database

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Magma 3c1d2230e6d6…

magma 3c1d2230e6d6
Size
961
Isomorphism class hash
3c1d2230e6d607f73abfa71c2eebd989d8b9366ea7b9f0ec007a81544de058b9
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
yes
Submitted by
b-reinke
Submitted at
2026-05-14 05:15:52
Display reorder
0,2,189,826,329,956,259,233,303,546,594,761,51,618,366,153,708,129,380,628,34,766,583,566,304,229,285,937,322,817,186,1,636,193,841,409,307,105,505,176,645,788,606,939,443,573,40,45,571,422,954,603,768,655,167,504,119,300,413,849,206,610,754,657,730,884,800,390,592,228,377,416,692,290,497,902,570,845,574,925,480,317,695,417,369,234,585,400,811,886,718,643,741,929,719,579,221,898,159,808,538,769,451,407,54,164,866,379,367,872,179,63,415,449,787,531,803,155,897,210,598,729,932,666,850,900,660,355,812,484,638,765,203,144,72,32,686,26,256,8,701,53,73,138,196,762,608,493,831,359,640,927,840,774,782,86,947,360,117,412,878,920,278,499,62,5,318,527,664,668,542,289,29,55,510,266,907,892,410,107,354,946,91,330,231,321,791,896,523,19,323,462,236,895,511,860,191,604,312,141,295,605,184,862,498,875,226,470,328,15,514,899,797,699,632,614,688,401,728,621,327,293,836,479,421,197,362,541,767,81,64,760,528,352,202,444,453,854,314,324,625,720,389,246,42,794,43,248,255,642,893,564,133,853,381,387,786,9,474,732,960,716,458,25,770,403,365,837,149,548,877,658,239,104,615,280,264,631,120,704,609,340,298,838,469,428,218,368,512,744,80,65,751,525,378,213,437,463,852,309,343,637,712,385,232,59,778,58,230,405,530,31,344,454,242,891,482,846,168,589,311,157,296,588,175,844,495,879,252,478,339,3,539,912,775,672,630,616,683,781,915,123,394,876,923,282,494,39,13,305,526,652,648,543,302,21,46,483,262,904,894,396,101,376,931,70,337,247,346,75,930,370,815,509,624,746,212,154,89,37,685,24,272,10,702,48,88,128,219,749,622,500,828,371,644,916,834,798,790,856,911,656,375,139,819,515,771,465,399,47,163,888,383,363,882,180,38,391,467,785,522,824,143,914,205,595,735,945,650,948,713,582,194,913,601,250,353,433,679,301,496,909,558,861,554,918,481,306,676,432,361,244,576,406,816,868,714,641,738,757,659,734,870,827,384,488,183,671,784,587,938,438,555,61,56,557,427,955,590,772,661,160,489,121,291,393,857,209,620,17,626,222,833,397,316,103,288,550,599,750,50,612,364,145,724,137,382,634,35,745,578,562,319,249,275,940,345,810,173,18,16,170,801,338,953,269,245,490,215,717,905,125,273,452,136,146,448,271,99,922,731,216,487,687,516,544,617,299,950,737,411,758,943,308,629,568,521,700,577,95,207,477,653,447,225,556,237,418,647,455,192,82,600,320,865,747,779,286,132,944,492,485,949,150,258,777,748,873,331,142,349,187,310,646,93,908,919,84,654,297,188,334,140,783,726,569,281,536,435,835,802,112,809,855,430,533,263,575,722,773,829,456,743,547,98,162,639,181,126,565,752,476,814,667,680,952,41,890,534,69,235,832,858,227,76,535,880,44,941,675,665,100,475,502,325,36,807,804,49,326,507,457,124,740,7,398,698,707,429,776,166,277,869,267,177,780,436,709,689,392,27,755,540,335,127,14,591,357,586,20,97,348,529,611,425,871,90,690,151,386,567,821,204,195,822,545,404,131,697,87,867,440,635,711,83,847,106,677,678,118,843,94,705,185,439,581,926,12,942,240,198,268,372,450,374,276,201,254,951,22,901,596,426,190,238,217,260,358,466,356,284,214,224,958,28,910,597,423,169,721,78,859,102,694,693,122,863,67,727,174,442,580,917,6,935,388,561,806,211,220,805,551,402,135,674,74,887,441,627,513,332,113,4,602,373,607,30,111,351,524,619,424,883,71,681,147,796,161,283,885,261,182,792,445,723,682,414,23,756,108,473,491,342,33,820,823,52,341,486,459,116,739,11,408,673,725,420,92,243,842,848,251,85,518,874,57,936,696,651,830,460,736,549,110,165,623,178,114,563,759,472,813,649,691,957,60,864,519,839,825,96,818,851,419,520,265,553,710,799,156,350,172,313,670,68,903,924,77,662,294,171,333,158,789,706,559,279,517,446,933,501,508,928,148,270,795,763,881,336,584,66,208,471,663,434,253,572,241,431,669,461,223,79,593,347,889,764,793,274,134,742,395,753,934,315,633,560,532,703,503,200,715,906,115,287,464,130,152,468,257,109,921,733,199,506,684,537,552,613,292,959 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 — but in a DIFFERENT family from the AG(2, 31) line construction (sibling magmas magma#2b8e3c08, magma#5f44e78d, magma#ba90950a, magma#fc322ba7 use that template). Distinguishing features: • <0, 1> has size 961 — 2 elements generate the whole magma, so there are no size-31 (= AG(2, 31) line) sub-magmas through 0. This rules out the AG(2, 31) line family. • L_0 = T[0, ·] has cycle structure (1, 60¹⁶): 16 orbits of length 60 plus the fixed point at 0. So L_0 has order 60 (vs. order 10 for the AG-line family). • The carrier is F_961 = GF(31²) additively (= (Z/31)² as elementary abelian 31²-group). Translation by 1 in this hidden structure is an order-31 fix-free magma automorphism (verified by extending {σ(0)=1, σ(1)=754}); commutativity then yields a second independent translation τ₂, giving the full (Z/31)² translation group. Structure: x ◇ y = x + f(y − x) on (Z/31)², where f has cycle structure (1, 60¹⁶) — 16 cosets of an order-60 subgroup ⟨c⟩ in F_961* (which has order 960 = 16 · 60). On each coset f acts as multiplication by some near-field element; the 16-way coset decomposition makes this a 16-coset 'Tao Type II'-style piecewise-linear construction on F_961. In the display-reorder coordinates, f is F_31-linear along each AXIS (f((c, 0)) = c · (8, 12) and f((0, d)) = d · (16, 8) mod 31) but NOT F_31-linear globally: f((1, 1)) = (23, 5) ≠ (24, 20) = f((1, 0)) + f((0, 1)). This axis-additivity-without-global-linearity is the structural fingerprint of NEAR-FIELD multiplication (compare magma#5ebfbb80 for the exceptional near-field of order 11², and magma#29114da6 for the exceptional near-field of order 29²). Since 31 is NOT in the Zassenhaus exceptional list {5, 7, 11, 23, 29, 59}, the near-field underlying this magma is presumably a Dickson near-field of order 31² with the order-2 Frobenius automorphism, with c of order 60 in the Dickson multiplicative group. Display reorder presents elements as (i, j) ↦ τ₁ⁱ τ₂ʲ(0) at index 31i + j, where τ₁, τ₂ are commuting order-31 fix-free magma automorphisms (the hidden additive translations in (Z/31)²). Under this reorder the Cayley table is fully (Z/31)²-translation-invariant. [text written by Claude]

last edited by dwrensha at 2026-05-14 21:34:37 · history