Summary

You searched for: sol=36

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1

New Number: 2.11 |  AESZ: 69  |  Superseeker: 64 246848  |  Hash: 729adc350de26d9415643078ed8d3867  

Degree: 2

\(\theta^4-2^{2} x(4\theta+1)(4\theta+3)(10\theta^2+10\theta+3)+2^{4} 3^{2} x^{2}(4\theta+1)(4\theta+3)(4\theta+5)(4\theta+7)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 36, 6300, 1718640, 575675100, ...
--> OEIS
Normalized instanton numbers (n0=1): 64, 2616, 246848, 32024824, 5160268864, ... ; Common denominator:...

Discriminant

\((576z-1)(64z-1)\)

Local exponents

\(0\)\(\frac{ 1}{ 576}\)\(\frac{ 1}{ 64}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 4}\)
\(0\)\(1\)\(1\)\(\frac{ 3}{ 4}\)
\(0\)\(1\)\(1\)\(\frac{ 5}{ 4}\)
\(0\)\(2\)\(2\)\(\frac{ 7}{ 4}\)

Note:

Hadamard product $C\ast c$

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2

New Number: 2.22 |  AESZ: 135  |  Superseeker: 36 -206716/3  |  Hash: 85e55291bd94bb32087b43f104c60645  

Degree: 2

\(\theta^4-2^{2} 3 x(4\theta+1)(4\theta+3)(3\theta^2+3\theta+1)+2^{4} 3^{3} x^{2}(4\theta+1)(4\theta+3)(4\theta+5)(4\theta+7)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 36, 3780, 388080, 8108100, ...
--> OEIS
Normalized instanton numbers (n0=1): 36, -477, -206716/3, -4431924, -27005472, ... ; Common denominator:...

Discriminant

\(1-576z+110592z^2\)

Local exponents

\(0\)\(\frac{ 1}{ 384}-\frac{ 1}{ 1152}\sqrt{ 3}I\)\(\frac{ 1}{ 384}+\frac{ 1}{ 1152}\sqrt{ 3}I\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 4}\)
\(0\)\(1\)\(1\)\(\frac{ 3}{ 4}\)
\(0\)\(1\)\(1\)\(\frac{ 5}{ 4}\)
\(0\)\(2\)\(2\)\(\frac{ 7}{ 4}\)

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3

New Number: 2.25 |  AESZ: 138  |  Superseeker: 27 2618  |  Hash: c524254b716132352b27914640b03c8b  

Degree: 2

\(\theta^4-3 x(3\theta+1)(3\theta+2)(17\theta^2+17\theta+6)+2^{3} 3^{4} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 36, 3780, 524160, 82952100, ...
--> OEIS
Normalized instanton numbers (n0=1): 27, 189/4, 2618, 43713, 2319057, ... ; Common denominator:...

Discriminant

\((243z-1)(216z-1)\)

Local exponents

\(0\)\(\frac{ 1}{ 243}\)\(\frac{ 1}{ 216}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 3}\)
\(0\)\(1\)\(1\)\(\frac{ 2}{ 3}\)
\(0\)\(1\)\(1\)\(\frac{ 4}{ 3}\)
\(0\)\(2\)\(2\)\(\frac{ 5}{ 3}\)

Note:

Hadamard product $B\ast g$.

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4

New Number: 2.7 |  AESZ: 51  |  Superseeker: 92 585396  |  Hash: e09b9b149b6845daa8d5ef03df33f22d  

Degree: 2

\(\theta^4-2^{2} x(4\theta+1)(4\theta+3)(11\theta^2+11\theta+3)-2^{4} x^{2}(4\theta+1)(4\theta+3)(4\theta+5)(4\theta+7)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 36, 7980, 2716560, 1127025900, ...
--> OEIS
Normalized instanton numbers (n0=1): 92, 5052, 585396, 99982012, 21054159152, ... ; Common denominator:...

Discriminant

\(1-704z-4096z^2\)

Local exponents

\(-\frac{ 11}{ 128}-\frac{ 5}{ 128}\sqrt{ 5}\)\(0\)\(-\frac{ 11}{ 128}+\frac{ 5}{ 128}\sqrt{ 5}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 4}\)
\(1\)\(0\)\(1\)\(\frac{ 3}{ 4}\)
\(1\)\(0\)\(1\)\(\frac{ 5}{ 4}\)
\(2\)\(0\)\(2\)\(\frac{ 7}{ 4}\)

Note:

Hadamard product C*b
Related to 8.139
A-Incarnation: double cover of $B_5$.

A:Incarnation: double cover of B

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5

New Number: 5.130 |  AESZ:  |  Superseeker: 108 122756  |  Hash: 829aca3d7a00547e299bf794c8643162  

Degree: 5

\(\theta^4-2^{2} 3 x\left(12\theta^4+96\theta^3+71\theta^2+23\theta+3\right)-2^{4} 3^{3} x^{2}\left(160\theta^4+64\theta^3-544\theta^2-340\theta-65\right)+2^{8} 3^{5} x^{3}\left(32\theta^4+576\theta^3+588\theta^2+240\theta+35\right)+2^{12} 3^{7} x^{4}(28\theta^2+52\theta+31)(2\theta+1)^2+2^{16} 3^{9} x^{5}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 36, 3780, 558000, 98828100, ...
--> OEIS
Normalized instanton numbers (n0=1): 108, -1782, 122756, -5930658, 607239072, ... ; Common denominator:...

Discriminant

\((144z-1)(6912z^2+288z-1)(1+144z)^2\)

Local exponents

\(-\frac{ 1}{ 48}-\frac{ 1}{ 72}\sqrt{ 3}\)\(-\frac{ 1}{ 144}\)\(0\)\(-\frac{ 1}{ 48}+\frac{ 1}{ 72}\sqrt{ 3}\)\(\frac{ 1}{ 144}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(3\)\(0\)\(1\)\(1\)\(\frac{ 3}{ 2}\)
\(2\)\(4\)\(0\)\(2\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.130" from ...

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6

New Number: 5.134 |  AESZ:  |  Superseeker: 176 1215248/3  |  Hash: 5bd656df18dc5d02b2f2a068ba88ab74  

Degree: 5

\(\theta^4+2^{2} x\left(4\theta^4-352\theta^3-250\theta^2-74\theta-9\right)-2^{4} 3 x^{2}\left(3168\theta^4+5952\theta^3-3712\theta^2-2648\theta-519\right)-2^{8} 3^{3} x^{3}\left(2912\theta^4-3008\theta^3-3152\theta^2-1160\theta-145\right)+2^{12} 3^{3} 5 x^{4}(2\theta+1)(824\theta^3+1668\theta^2+1342\theta+405)+2^{16} 3^{4} 5^{2} x^{5}(2\theta+1)(6\theta+5)(6\theta+7)(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 36, 4572, 918000, 228519900, ...
--> OEIS
Normalized instanton numbers (n0=1): 176, -3238, 1215248/3, -18807038, 3651829680, ... ; Common denominator:...

Discriminant

\((48z-1)(432z-1)(16z+1)(1+240z)^2\)

Local exponents

\(-\frac{ 1}{ 16}\)\(-\frac{ 1}{ 240}\)\(0\)\(\frac{ 1}{ 432}\)\(\frac{ 1}{ 48}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(\frac{ 5}{ 6}\)
\(1\)\(3\)\(0\)\(1\)\(1\)\(\frac{ 7}{ 6}\)
\(2\)\(4\)\(0\)\(2\)\(2\)\(\frac{ 3}{ 2}\)

Note:

B-incarnation as fibre product 61131- x 182--1

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7

New Number: 5.65 |  AESZ: 273  |  Superseeker: 63/5 14016/5  |  Hash: cf49bc645cb0404ce7bc9ca1d41d3152  

Degree: 5

\(5^{2} \theta^4-3 5 x\left(333\theta^4+882\theta^3+781\theta^2+340\theta+60\right)+2^{2} 3^{2} x^{2}\left(5184\theta^4+40662\theta^3+71829\theta^2+47700\theta+11540\right)+2^{2} 3^{5} x^{3}\left(8424\theta^4+9720\theta^3-46899\theta^2-63045\theta-21260\right)-2^{4} 3^{8} x^{4}\left(1296\theta^4+12312\theta^3+23094\theta^2+16263\theta+3956\right)-2^{6} 3^{11} x^{5}(3\theta+2)(3\theta+4)(6\theta+5)(6\theta+7)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 36, 2196, 161280, 13032900, ...
--> OEIS
Normalized instanton numbers (n0=1): 63/5, -747/4, 14016/5, -55584, 6071598/5, ... ; Common denominator:...

Discriminant

\(-(-1+27z)(108z+5)^2(108z-1)^2\)

Local exponents

\(-\frac{ 5}{ 108}\)\(0\)\(\frac{ 1}{ 108}\)\(\frac{ 1}{ 27}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 2}{ 3}\)
\(1\)\(0\)\(\frac{ 1}{ 6}\)\(1\)\(\frac{ 5}{ 6}\)
\(3\)\(0\)\(\frac{ 5}{ 6}\)\(1\)\(\frac{ 7}{ 6}\)
\(4\)\(0\)\(1\)\(2\)\(\frac{ 4}{ 3}\)

Note:

This is operator "5.65" from ...

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8

New Number: 13.9 |  AESZ:  |  Superseeker: 8 2200/9  |  Hash: 31ff3b7bd4c8fed070ee43b6903d3752  

Degree: 13

\(\theta^4+2^{3} x\theta(4\theta^3-8\theta^2-5\theta-1)-2^{4} x^{2}\left(48\theta^4+120\theta^3+45\theta^2+74\theta+36\right)-2^{7} x^{3}\left(101\theta^4-342\theta^3-387\theta^2-410\theta-171\right)+2^{8} x^{4}\left(3121\theta^4+14104\theta^3+30889\theta^2+27720\theta+9351\right)+2^{11} 3^{2} x^{5}\left(655\theta^4+4062\theta^3+10081\theta^2+10272\theta+3856\right)-2^{12} 3^{2} x^{6}\left(2272\theta^4+2816\theta^3-9950\theta^2-18768\theta-8813\right)-2^{15} 3^{3} x^{7}\left(1546\theta^4+12172\theta^3+30708\theta^2+33880\theta+13843\right)+2^{16} 3^{4} x^{8}\left(1099\theta^4+1344\theta^3-11134\theta^2-23964\theta-13063\right)+2^{19} 3^{5} x^{9}\left(458\theta^4+4828\theta^3+15325\theta^2+19721\theta+8830\right)-2^{20} 3^{6} x^{10}(\theta+1)(368\theta^3+1752\theta^2+1297\theta-1035)-2^{23} 3^{7} x^{11}(\theta+1)(\theta+2)(39\theta^2+513\theta+1172)+2^{24} 3^{9} x^{12}(\theta+3)(\theta+2)(\theta+1)(17\theta+82)-2^{27} 3^{10} x^{13}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 0, 36, -192, -4284, ...
--> OEIS
Normalized instanton numbers (n0=1): 8, -75/2, 2200/9, -8117/2, 47936, ... ; Common denominator:...

Discriminant

\(-(8z+1)(5184z^3+432z^2-36z+1)(12z+1)^2(144z^2-24z-1)^2(4z-1)^3\)

Local exponents

≈\(-0.141868\)\(-\frac{ 1}{ 8}\)\(-\frac{ 1}{ 12}\)\(\frac{ 1}{ 12}-\frac{ 1}{ 12}\sqrt{ 2}\)\(0\) ≈\(0.029267-0.022431I\) ≈\(0.029267+0.022431I\)\(\frac{ 1}{ 12}+\frac{ 1}{ 12}\sqrt{ 2}\)\(\frac{ 1}{ 4}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(2\)
\(1\)\(1\)\(\frac{ 1}{ 2}\)\(3\)\(0\)\(1\)\(1\)\(3\)\(\frac{ 3}{ 2}\)\(3\)
\(2\)\(2\)\(1\)\(4\)\(0\)\(2\)\(2\)\(4\)\(2\)\(4\)

Note:

This is operator "13.9" from ...

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9

New Number: 14.8 |  AESZ:  |  Superseeker: 92/5 -76/5  |  Hash: a787adbb87527c14af9a5f2508991317  

Degree: 14

\(5^{2} \theta^4-2^{2} 5 x\left(244\theta^4+496\theta^3+473\theta^2+225\theta+45\right)+2^{4} x^{2}\left(24144\theta^4+95872\theta^3+155244\theta^2+117660\theta+36835\right)-2^{9} x^{3}\left(33248\theta^4+180880\theta^3+407234\theta^2+416430\theta+168275\right)+2^{14} x^{4}\left(32428\theta^4+181000\theta^3+474021\theta^2+605903\theta+294817\right)-2^{19} x^{5}\left(29162\theta^4+130236\theta^3+270865\theta^2+378135\theta+208235\right)+2^{24} x^{6}\left(25019\theta^4+101558\theta^3+110864\theta^2+69739\theta+20552\right)-2^{29} x^{7}\left(17583\theta^4+76998\theta^3+91248\theta^2+4717\theta-39868\right)+2^{34} x^{8}\left(10081\theta^4+40990\theta^3+72600\theta^2+35261\theta-9816\right)-2^{39} x^{9}\left(5407\theta^4+16538\theta^3+33406\theta^2+27573\theta+6100\right)+2^{44} x^{10}\left(2699\theta^4+7862\theta^3+13380\theta^2+11845\theta+4325\right)-2^{49} x^{11}\left(1033\theta^4+3966\theta^3+6998\theta^2+6213\theta+2329\right)+2^{54} x^{12}\left(255\theta^4+1318\theta^3+2815\theta^2+2864\theta+1159\right)-2^{59} x^{13}\left(35\theta^4+230\theta^3+589\theta^2+692\theta+313\right)+2^{65} x^{14}\left((\theta+2)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 36, 1196, 41488, 1543916, ...
--> OEIS
Normalized instanton numbers (n0=1): 92/5, -342/5, -76/5, 75394/5, -2156752/5, ... ; Common denominator:...

Discriminant

\((64z-1)(32z-1)(256z^2-48z+1)(32768z^3-1024z^2-5-32z)^2(16z-1)^4\)

Local exponents

≈\(-0.020941-0.040594I\) ≈\(-0.020941+0.040594I\)\(0\)\(\frac{ 1}{ 64}\)\(\frac{ 3}{ 32}-\frac{ 1}{ 32}\sqrt{ 5}\)\(\frac{ 1}{ 32}\)\(\frac{ 1}{ 16}\) ≈\(0.073133\)\(\frac{ 3}{ 32}+\frac{ 1}{ 32}\sqrt{ 5}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(-\frac{ 1}{ 2}\)\(0\)\(0\)\(2\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(-\frac{ 1}{ 2}\)\(1\)\(1\)\(2\)
\(3\)\(3\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(3\)\(1\)\(2\)
\(4\)\(4\)\(0\)\(2\)\(2\)\(2\)\(\frac{ 1}{ 2}\)\(4\)\(2\)\(2\)

Note:

This is operator "14.8" from ...

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10

New Number: 8.25 |  AESZ: 299  |  Superseeker: -54 -197216/3  |  Hash: c9e3907e21d64cf5564bf2d00992459e  

Degree: 8

\(\theta^4-2 3 x\left(144\theta^4+36\theta^3+47\theta^2+29\theta+6\right)+2^{2} 3^{2} x^{2}\left(8376\theta^4+6648\theta^3+8157\theta^2+3900\theta+724\right)-2^{4} 3^{4} x^{3}\left(42672\theta^4+68616\theta^3+81056\theta^2+44841\theta+9964\right)+2^{6} 3^{5} x^{4}\left(374028\theta^4+962040\theta^3+1262091\theta^2+794463\theta+195335\right)-2^{8} 3^{7} x^{5}\left(633840\theta^4+2243328\theta^3+3405968\theta^2+2385208\theta+629129\right)+2^{12} 3^{8} x^{6}\left(438960\theta^4+1884384\theta^3+3176664\theta^2+2380392\theta+652943\right)-2^{19} 3^{10} x^{7}\left(5760\theta^4+25128\theta^3+39548\theta^2+26606\theta+6517\right)+2^{22} 3^{11} x^{8}(6\theta+5)^2(6\theta+7)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 36, 1908, 116496, 7816500, ...
--> OEIS
Normalized instanton numbers (n0=1): -54, -1530, -197216/3, -3553920, -222887448, ... ; Common denominator:...

Discriminant

\((1-144z+6912z^2)(108z-1)^2(3456z^2-252z+1)^2\)

Local exponents

\(0\)\(\frac{ 7}{ 192}-\frac{ 1}{ 576}\sqrt{ 345}\)\(\frac{ 1}{ 108}\)\(\frac{ 1}{ 96}-\frac{ 1}{ 288}\sqrt{ 3}I\)\(\frac{ 1}{ 96}+\frac{ 1}{ 288}\sqrt{ 3}I\)\(\frac{ 7}{ 192}+\frac{ 1}{ 576}\sqrt{ 345}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 5}{ 6}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)\(\frac{ 5}{ 6}\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(3\)\(\frac{ 7}{ 6}\)
\(0\)\(4\)\(1\)\(2\)\(2\)\(4\)\(\frac{ 7}{ 6}\)

Note:

This is operator "8.25" from ...

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11

New Number: 8.78 |  AESZ:  |  Superseeker: 52 48732  |  Hash: 2fb524ad6efb19e0117ae7acbd9f67b9  

Degree: 8

\(\theta^4-2^{2} x\left(184\theta^4+224\theta^3+175\theta^2+63\theta+9\right)+2^{4} 3 x^{2}\left(3472\theta^4+9664\theta^3+9864\theta^2+4264\theta+705\right)-2^{8} 3^{2} x^{3}\left(1936\theta^4+27936\theta^3+43336\theta^2+21528\theta+3933\right)-2^{16} 3^{3} x^{4}\left(1384\theta^4+524\theta^3-4555\theta^2-3404\theta-753\right)+2^{19} 3^{4} x^{5}\left(3440\theta^4+13712\theta^3-58\theta^2-3774\theta-1161\right)+2^{22} 3^{5} x^{6}\left(11312\theta^4-9888\theta^3-10808\theta^2-1608\theta+459\right)-2^{26} 3^{7} x^{7}(2\theta+1)(1336\theta^3+2772\theta^2+2234\theta+663)-2^{32} 3^{9} x^{8}(2\theta+1)(4\theta+3)(4\theta+5)(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 36, 3780, 555120, 95199300, ...
--> OEIS
Normalized instanton numbers (n0=1): 52, -399, 48732, -992750, 98106208, ... ; Common denominator:...

Discriminant

\(-(256z-1)(110592z^3+6912z^2-288z+1)(-1+96z+13824z^2)^2\)

Local exponents

≈\(-0.091906\)\(-\frac{ 1}{ 288}-\frac{ 1}{ 288}\sqrt{ 7}\)\(0\) ≈\(0.00385\)\(\frac{ 1}{ 256}\)\(-\frac{ 1}{ 288}+\frac{ 1}{ 288}\sqrt{ 7}\) ≈\(0.025556\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)\(\frac{ 3}{ 4}\)
\(1\)\(3\)\(0\)\(1\)\(1\)\(3\)\(1\)\(\frac{ 5}{ 4}\)
\(2\)\(4\)\(0\)\(2\)\(2\)\(4\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "8.78" from ...

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12

New Number: 1.4 |  AESZ: 4  |  Superseeker: 117 713814  |  Hash: 1f2a9672b7cdc68eae658b2304b40dbd  

Degree: 1

\(\theta^4-3^{2} x(3\theta+1)^2(3\theta+2)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 36, 8100, 2822400, 1200622500, ...
--> OEIS
Normalized instanton numbers (n0=1): 117, 5868, 713814, 126605376, 27754210287, ... ; Common denominator:...

Discriminant

\(\)

No data for singularities

Note:

A-incarnation: X(3,3) in P^5.

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