Calabi-Yau differential operator database v.3.0 - Search results

Summary

You searched for: Spectrum0=0,1,1,2

Your search produced 482 matches
1-30  31-60  61-90  91-120  121-150  151-180
181-210  211-240  241-270  271-300  301-330  331-360
361-390  391-420  421-450  451-480  481-482

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361

New Number: 7.1 | AESZ: | Superseeker: 10/7 508/7 | Hash: 08ab3cb496250adfa30bc3e24ac63c4f

Degree: 7

\(7^{2} \theta^4-2 7 x\theta(46\theta^3+52\theta^2+33\theta+7)-2^{2} x^{2}\left(7332\theta^4+28848\theta^3+42633\theta^2+26670\theta+6272\right)-2^{4} x^{3}\left(2860\theta^4+44760\theta^3+120483\theta^2+111279\theta+35098\right)+2^{9} x^{4}\left(2230\theta^4+5920\theta^3-741\theta^2-6509\theta-3049\right)+2^{14} x^{5}\left(174\theta^4+1320\theta^3+1971\theta^2+1095\theta+190\right)-2^{19} x^{6}\left(22\theta^4+24\theta^3-9\theta^2-21\theta-7\right)-2^{25} x^{7}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 0, 32, 288, 7776, ...
--> OEIS
Normalized instanton numbers (n₀=1): 10/7, 100/7, 508/7, 808, 59910/7, ... ; Common denominator:...

Discriminant

\(-(16z+1)(32z-1)(32z-7)^2(4z+1)^3\)

Local exponents

\(-\frac{ 1}{ 4}\) \(-\frac{ 1}{ 16}\) \(0\) \(\frac{ 1}{ 32}\) \(\frac{ 7}{ 32}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(\frac{ 1}{ 2}\) \(1\) \(0\) \(1\) \(1\) \(1\)
\(\frac{ 3}{ 2}\) \(1\) \(0\) \(1\) \(3\) \(1\)
\(2\) \(2\) \(0\) \(2\) \(4\) \(1\)

Note:

This is operator "7.1" from ...

362

New Number: 7.20 | AESZ: | Superseeker: 10 3394/3 | Hash: 9d5791eaabb9d0e9cb4b5cd0b2158b12

Degree: 7

\(\theta^4-x\left(88\theta^3-4+71\theta^4+42\theta^2-2\theta\right)-x^{2}\left(10462\theta+13294\theta^2+875\theta^4+6848\theta^3+3132\right)+3^{2} x^{3}\left(373\theta^4-6360\theta^3-30716\theta^2-44868\theta-23180\right)+3^{4} x^{4}\left(1843\theta^4+8384\theta^3+3236\theta^2-14996\theta-15180\right)+3^{8} x^{5}\left(75\theta^4+1272\theta^3+3454\theta^2+3554\theta+1192\right)-3^{11} x^{6}\left(27\theta^4-414\theta^2-918\theta-584\right)-3^{16} x^{7}\left((\theta+2)^4\right)\)

Coefficients of the holomorphic solution: 1, -4, 147, 4496, 223111, ...
--> OEIS
Normalized instanton numbers (n₀=1): 10, 77, 3394/3, 24029, 640402, ... ; Common denominator:...

Discriminant

\(-(-1+81z)(9z-1)^2(81z^2+14z+1)^2\)

Local exponents

\(-\frac{ 7}{ 81}-\frac{ 4}{ 81}\sqrt{ 2}I\) \(-\frac{ 7}{ 81}+\frac{ 4}{ 81}\sqrt{ 2}I\) \(0\) \(\frac{ 1}{ 81}\) \(\frac{ 1}{ 9}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(2\)
\(-\frac{ 1}{ 2}\) \(-\frac{ 1}{ 2}\) \(0\) \(1\) \(1\) \(2\)
\(1\) \(1\) \(0\) \(1\) \(3\) \(2\)
\(\frac{ 3}{ 2}\) \(\frac{ 3}{ 2}\) \(0\) \(2\) \(4\) \(2\)

Note:

This is operator "7.20" from ...

363

New Number: 7.21 | AESZ: | Superseeker: 90 413926 | Hash: f2cdf32038c22a3da2f5752ad59eaa27

Degree: 7

\(\theta^4-3^{2} x\left(27\theta^4+216\theta^3+234\theta^2+126\theta+28\right)-3^{6} x^{2}\left(75\theta^4-672\theta^3-2378\theta^2-2602\theta-1076\right)+3^{9} x^{3}\left(1843\theta^4+6360\theta^3-2836\theta^2-13692\theta-9828\right)-3^{14} x^{4}\left(373\theta^4+9344\theta^3+16396\theta^2+10260\theta+540\right)-3^{19} x^{5}\left(875\theta^4+152\theta^3-6794\theta^2-11462\theta-5400\right)+3^{26} x^{6}\left(71\theta^4+480\theta^3+1218\theta^2+1386\theta+600\right)+3^{33} x^{7}\left((\theta+2)^4\right)\)

Coefficients of the holomorphic solution: 1, 252, 40419, 2460816, -1025424441, ...
--> OEIS
Normalized instanton numbers (n₀=1): 90, -4365, 413926, -38862153, 4502063682, ... ; Common denominator:...

Discriminant

\((1+27z)(243z+1)^2(59049z^2-378z+1)^2\)

Local exponents

\(-\frac{ 1}{ 27}\) \(-\frac{ 1}{ 243}\) \(0\) \(\frac{ 7}{ 2187}-\frac{ 4}{ 2187}\sqrt{ 2}I\) \(\frac{ 7}{ 2187}+\frac{ 4}{ 2187}\sqrt{ 2}I\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(2\)
\(1\) \(1\) \(0\) \(-\frac{ 1}{ 2}\) \(-\frac{ 1}{ 2}\) \(2\)
\(1\) \(3\) \(0\) \(1\) \(1\) \(2\)
\(2\) \(4\) \(0\) \(\frac{ 3}{ 2}\) \(\frac{ 3}{ 2}\) \(2\)

Note:

This is operator "7.21" from ...

364

New Number: 7.2 | AESZ: | Superseeker: -80 -249872 | Hash: 341389ebf4ab0242c5b70d9a8fd7a1d9

Degree: 7

\(\theta^4+2^{4} x\left(22\theta^4+64\theta^3+51\theta^2+19\theta+3\right)-2^{9} x^{2}\left(174\theta^4-624\theta^3-945\theta^2-417\theta-80\right)-2^{14} x^{3}\left(2230\theta^4+3000\theta^3-5121\theta^2-3813\theta-971\right)+2^{19} x^{4}\left(2860\theta^4-33320\theta^3+3363\theta^2+6847\theta+2402\right)+2^{27} x^{5}\left(7332\theta^4+480\theta^3+81\theta^2+1380\theta+719\right)+2^{36} 7 x^{6}(\theta+1)(46\theta^3+86\theta^2+67\theta+20)-2^{45} 7^{2} x^{7}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, -48, 5072, -733440, 124117776, ...
--> OEIS
Normalized instanton numbers (n₀=1): -80, -4202, -249872, -22251117, -2195810928, ... ; Common denominator:...

Discriminant

\(-(64z+1)(32z-1)(224z-1)^2(256z+1)^3\)

Local exponents

\(-\frac{ 1}{ 64}\) \(-\frac{ 1}{ 256}\) \(0\) \(\frac{ 1}{ 224}\) \(\frac{ 1}{ 32}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(\frac{ 1}{ 2}\) \(0\) \(1\) \(1\) \(1\)
\(1\) \(\frac{ 3}{ 2}\) \(0\) \(3\) \(1\) \(1\)
\(2\) \(2\) \(0\) \(4\) \(2\) \(1\)

Note:

This is operator "7.2" from ...

365

New Number: 7.3 | AESZ: | Superseeker: 3 64 | Hash: 8413250555ca536f1bdccfeed506ea4e

Degree: 7

\(\theta^4+x\theta(39\theta^3-30\theta^2-19\theta-4)+2 x^{2}\left(16\theta^4-1070\theta^3-1057\theta^2-676\theta-192\right)-2^{2} 3^{2} x^{3}(3\theta+2)(171\theta^3+566\theta^2+600\theta+316)-2^{5} 3^{3} x^{4}\left(384\theta^4+1542\theta^3+2635\theta^2+2173\theta+702\right)-2^{6} 3^{3} x^{5}(\theta+1)(1393\theta^3+5571\theta^2+8378\theta+4584)-2^{10} 3^{5} x^{6}(\theta+1)(\theta+2)(31\theta^2+105\theta+98)-2^{12} 3^{7} x^{7}(\theta+1)(\theta+2)^2(\theta+3)\)

Coefficients of the holomorphic solution: 1, 0, 24, 192, 3384, ...
--> OEIS
Normalized instanton numbers (n₀=1): 3, -4, 64, -253, 4292, ... ; Common denominator:...

Discriminant

\(-(8z+1)(24z-1)(3z+1)(4z+1)(12z+1)(1+18z)^2\)

Local exponents

\(-\frac{ 1}{ 3}\) \(-\frac{ 1}{ 4}\) \(-\frac{ 1}{ 8}\) \(-\frac{ 1}{ 12}\) \(-\frac{ 1}{ 18}\) \(0\) \(\frac{ 1}{ 24}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(1\) \(1\) \(1\) \(0\) \(1\) \(2\)
\(1\) \(1\) \(1\) \(1\) \(3\) \(0\) \(1\) \(2\)
\(2\) \(2\) \(2\) \(2\) \(4\) \(0\) \(2\) \(3\)

Note:

This is operator "7.3" from ...

366

New Number: 7.4 | AESZ: | Superseeker: -988/3 -14008436/3 | Hash: 9bddfb88498c0a263be6ca541ae7e980

Degree: 7

\(3^{2} \theta^4+2^{2} 3 x\left(760\theta^4+2048\theta^3+1423\theta^2+399\theta+42\right)-2^{7} x^{2}\left(20440\theta^4+25216\theta^3-4415\theta^2-4845\theta-795\right)+2^{12} x^{3}\left(39928\theta^4+16512\theta^3+23719\theta^2+11637\theta+1830\right)+2^{17} x^{4}\left(2928\theta^4-41856\theta^3-42871\theta^2-16873\theta-2425\right)+2^{23} x^{5}\left(608\theta^4+3968\theta^3+10676\theta^2+6177\theta+1089\right)+2^{29} x^{6}\left(272\theta^4+1056\theta^3+861\theta^2+264\theta+27\right)+2^{35} x^{7}\left((2\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, -56, 21096, -12540800, 9146271400, ...
--> OEIS
Normalized instanton numbers (n₀=1): -988/3, 57289/3, -14008436/3, 1385404666, -1599785191904/3, ... ; Common denominator:...

Discriminant

\((1+1248z-10240z^2+131072z^3)(-3+352z+2048z^2)^2\)

Local exponents

\(-\frac{ 11}{ 128}-\frac{ 1}{ 128}\sqrt{ 145}\) ≈\(-0.000796\) \(0\) \(-\frac{ 11}{ 128}+\frac{ 1}{ 128}\sqrt{ 145}\) ≈\(0.039461-0.089595I\) ≈\(0.039461+0.089595I\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(\frac{ 1}{ 2}\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(\frac{ 1}{ 2}\)
\(3\) \(1\) \(0\) \(3\) \(1\) \(1\) \(\frac{ 1}{ 2}\)
\(4\) \(2\) \(0\) \(4\) \(2\) \(2\) \(\frac{ 1}{ 2}\)

Note:

This is operator "7.4" from ...

367

New Number: 7.5 | AESZ: | Superseeker: 8096 9215266592 | Hash: 90403d92f72b2839164c2bbd30933deb

Degree: 7

\(\theta^4+2^{4} x\left(1088\theta^4-2048\theta^3-1260\theta^2-236\theta-19\right)+2^{15} x^{2}\left(1216\theta^4-5504\theta^3+11272\theta^2+3654\theta+423\right)+2^{24} x^{3}\left(11712\theta^4+190848\theta^3+97220\theta^2+27432\theta+2835\right)+2^{35} x^{4}\left(159712\theta^4+253376\theta^3+235372\theta^2+78648\theta+9491\right)-2^{46} x^{5}\left(81760\theta^4+62656\theta^3-46316\theta^2-33048\theta-5403\right)+2^{57} 3 x^{6}\left(3040\theta^4-2112\theta^3-2036\theta^2-528\theta-41\right)+2^{69} 3^{2} x^{7}\left((2\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 304, -113904, -2048902400, -4778502402800, ...
--> OEIS
Normalized instanton numbers (n₀=1): 8096, -9179600, 9215266592, -8060820053720, 27014124083677664, ... ; Common denominator:...

Discriminant

\((2147483648z^3+40894464z^2-5120z+1)(-1-11264z+6291456z^2)^2\)

Local exponents

≈\(-0.019169\) \(\frac{ 11}{ 12288}-\frac{ 1}{ 12288}\sqrt{ 145}\) \(0\) ≈\(6.3e-05-0.000143I\) ≈\(6.3e-05+0.000143I\) \(\frac{ 11}{ 12288}+\frac{ 1}{ 12288}\sqrt{ 145}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(\frac{ 1}{ 2}\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(\frac{ 1}{ 2}\)
\(1\) \(3\) \(0\) \(1\) \(1\) \(3\) \(\frac{ 1}{ 2}\)
\(2\) \(4\) \(0\) \(2\) \(2\) \(4\) \(\frac{ 1}{ 2}\)

Note:

This is operator "7.5" from ...

368

New Number: 7.6 | AESZ: | Superseeker: 1/3 5/3 | Hash: 24ba77c97bc4b46c39a41c77cc1d1ef4

Degree: 7

\(3^{2} \theta^4-3 x\left(112\theta^4+140\theta^3+133\theta^2+63\theta+12\right)+x^{2}\left(4393\theta^4+9340\theta^3+10903\theta^2+6360\theta+1488\right)-2 x^{3}\left(11669\theta^4+27720\theta^3+27019\theta^2+8460\theta-912\right)+2^{2} x^{4}\left(6799\theta^4-10288\theta^3-82183\theta^2-119168\theta-52672\right)+2^{3} 7 x^{5}(\theta+1)(2611\theta^3+15537\theta^2+26998\theta+14360)-2^{6} 7^{2} x^{6}(\theta+1)(\theta+2)(83\theta^2+105\theta-66)-2^{10} 7^{3} x^{7}(\theta+1)(\theta+2)^2(\theta+3)\)

Coefficients of the holomorphic solution: 1, 4, 28, 232, 2188, ...
--> OEIS
Normalized instanton numbers (n₀=1): 1/3, 5/6, 5/3, 19/3, 29, ... ; Common denominator:...

Discriminant

\(-(2z+1)(8z-1)(7z-1)(16z-1)(z+1)(-3+14z)^2\)

Local exponents

\(-1\) \(-\frac{ 1}{ 2}\) \(0\) \(\frac{ 1}{ 16}\) \(\frac{ 1}{ 8}\) \(\frac{ 1}{ 7}\) \(\frac{ 3}{ 14}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(2\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(3\) \(2\)
\(2\) \(2\) \(0\) \(2\) \(2\) \(2\) \(4\) \(3\)

Note:

This is operator "7.6" from ...

369

New Number: 7.7 | AESZ: | Superseeker: 2/3 13/3 | Hash: c7abb9c42d46f14955f0f23351082bef

Degree: 7

\(3^{2} \theta^4-2 3 x\left(88\theta^4+110\theta^3+103\theta^2+48\theta+9\right)+2^{2} x^{2}\left(2923\theta^4+6610\theta^3+8041\theta^2+4908\theta+1206\right)-x^{3}\left(123365\theta^4+374814\theta^3+519741\theta^2+346176\theta+89676\right)+2 x^{4}\left(309657\theta^4+1102938\theta^3+1591157\theta^2+1032920\theta+249740\right)-2^{3} 11 x^{5}(\theta+1)(12897\theta^3+35469\theta^2+31181\theta+8042)-2^{3} 11^{2} x^{6}(\theta+1)(\theta+2)(355\theta^2+1047\theta+806)-2^{4} 11^{3} x^{7}(\theta+1)(\theta+2)^2(\theta+3)\)

Coefficients of the holomorphic solution: 1, 6, 56, 636, 8196, ...
--> OEIS
Normalized instanton numbers (n₀=1): 2/3, 5/3, 13/3, 59/3, 119, ... ; Common denominator:...

Discriminant

\(-(11z-1)(4z^2+22z-1)(z^2+11z-1)(-3+22z)^2\)

Local exponents

\(-\frac{ 11}{ 2}-\frac{ 5}{ 2}\sqrt{ 5}\) \(-\frac{ 11}{ 4}-\frac{ 5}{ 4}\sqrt{ 5}\) \(0\) \(-\frac{ 11}{ 4}+\frac{ 5}{ 4}\sqrt{ 5}\) \(-\frac{ 11}{ 2}+\frac{ 5}{ 2}\sqrt{ 5}\) \(\frac{ 1}{ 11}\) \(\frac{ 3}{ 22}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(2\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(3\) \(2\)
\(2\) \(2\) \(0\) \(2\) \(2\) \(2\) \(4\) \(3\)

Note:

This is operator "7.7" from ...

370

New Number: 7.8 | AESZ: | Superseeker: -1/3 -5/3 | Hash: d5b8cfd5049e5d8670dac5bb5499d46a

Degree: 7

\(3^{2} \theta^4-3 x\left(272\theta^4+340\theta^3+347\theta^2+177\theta+36\right)+x^{2}\left(31273\theta^4+76540\theta^3+103783\theta^2+71112\theta+19728\right)-2 x^{3}\left(328219\theta^4+1181160\theta^3+1977957\theta^2+1620036\theta+522288\right)+2^{2} x^{4}\left(2036999\theta^4+9602752\theta^3+19022113\theta^2+17726192\theta+6309408\right)-2^{3} 17 x^{5}(\theta+1)(439669\theta^3+2114103\theta^2+3708554\theta+2306280)+2^{6} 3^{3} 17^{2} x^{6}(\theta+1)(\theta+2)(481\theta^2+1875\theta+1962)-2^{10} 3^{4} 17^{3} x^{7}(\theta+1)(\theta+2)^2(\theta+3)\)

Coefficients of the holomorphic solution: 1, 12, 156, 2136, 30348, ...
--> OEIS
Normalized instanton numbers (n₀=1): -1/3, 11/12, -5/3, 19/3, -29, ... ; Common denominator:...

Discriminant

\(-(17z-1)(9z-1)(8z-1)(18z-1)(16z-1)(-3+34z)^2\)

Local exponents

\(0\) \(\frac{ 1}{ 18}\) \(\frac{ 1}{ 17}\) \(\frac{ 1}{ 16}\) \(\frac{ 3}{ 34}\) \(\frac{ 1}{ 9}\) \(\frac{ 1}{ 8}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(0\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(2\)
\(0\) \(1\) \(1\) \(1\) \(3\) \(1\) \(1\) \(2\)
\(0\) \(2\) \(2\) \(2\) \(4\) \(2\) \(2\) \(3\)

Note:

This is operator "7.8" from ...

371

New Number: 8.10 | AESZ: 123 | Superseeker: 12 1828/3 | Hash: f0d76ab2b6b8808f4faa4ab8ecadff2c

Degree: 8

\(\theta^4-2^{2} x(3\theta^2+3\theta+1)(10\theta^2+10\theta+3)+2^{4} x^{2}\left(209\theta^4+1052\theta^3+1471\theta^2+838\theta+183\right)+2^{7} 3^{2} x^{3}\left(30\theta^4-180\theta^3-551\theta^2-417\theta-111\right)-2^{10} 3^{2} x^{4}\left(227\theta^4+454\theta^3-550\theta^2-777\theta-261\right)+2^{12} 3^{4} x^{5}\left(30\theta^4+300\theta^3+169\theta^2-25\theta-35\right)+2^{14} 3^{4} x^{6}\left(209\theta^4-216\theta^3-431\theta^2-216\theta-27\right)-2^{17} 3^{6} x^{7}(3\theta^2+3\theta+1)(10\theta^2+10\theta+3)+2^{20} 3^{8} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 12, 300, 10416, 431964, ...
--> OEIS
Normalized instanton numbers (n₀=1): 12, -47/2, 1828/3, -10813/4, 127948, ... ; Common denominator:...

Discriminant

\((36z-1)(8z-1)(72z-1)(4z-1)(-1+288z^2)^2\)

Local exponents

\(-\frac{ 1}{ 24}\sqrt{ 2}\) \(0\) \(\frac{ 1}{ 72}\) \(\frac{ 1}{ 36}\) \(\frac{ 1}{ 24}\sqrt{ 2}\) \(\frac{ 1}{ 8}\) \(\frac{ 1}{ 4}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(3\) \(0\) \(1\) \(1\) \(3\) \(1\) \(1\) \(1\)
\(4\) \(0\) \(2\) \(2\) \(4\) \(2\) \(2\) \(1\)

Note:

Hadamard product $c \ast d$. This operator has a second MUM-point at infinity with the same instanton numbers. It
can be reduced to an operator of degree 4 with a single
MUM-point defined over $\Q(\sqrt{?})$.

372

New Number: 8.11 | AESZ: 162 | Superseeker: 9 242/3 | Hash: 542708b59b898c35f43e00120897ff8d

Degree: 8

\(\theta^4-3 x(3\theta^2+3\theta+1)(10\theta^2+10\theta+3)+3^{3} x^{2}\left(91\theta^4+472\theta^3+659\theta^2+374\theta+81\right)+3^{6} x^{3}\left(30\theta^4-180\theta^3-551\theta^2-417\theta-111\right)-3^{8} x^{4}\left(200\theta^4+400\theta^3-514\theta^2-714\theta-237\right)+3^{11} x^{5}\left(30\theta^4+300\theta^3+169\theta^2-25\theta-35\right)+3^{13} x^{6}\left(91\theta^4-108\theta^3-211\theta^2-108\theta-15\right)-3^{16} x^{7}(3\theta^2+3\theta+1)(10\theta^2+10\theta+3)+3^{20} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 9, 135, 1953, 5751, ...
--> OEIS
Normalized instanton numbers (n₀=1): 9, -153/4, 242/3, -4923, 34245, ... ; Common denominator:...

Discriminant

\((27z^2-9z+1)(2187z^2-81z+1)(-1+243z^2)^2\)

Local exponents

\(-\frac{ 1}{ 27}\sqrt{ 3}\) \(0\) \(\frac{ 1}{ 54}-\frac{ 1}{ 162}\sqrt{ 3}I\) \(\frac{ 1}{ 54}+\frac{ 1}{ 162}\sqrt{ 3}I\) \(\frac{ 1}{ 27}\sqrt{ 3}\) \(\frac{ 1}{ 6}-\frac{ 1}{ 18}\sqrt{ 3}I\) \(\frac{ 1}{ 6}+\frac{ 1}{ 18}\sqrt{ 3}I\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(3\) \(0\) \(1\) \(1\) \(3\) \(1\) \(1\) \(1\)
\(4\) \(0\) \(2\) \(2\) \(4\) \(2\) \(2\) \(1\)

Note:

Hadamard product $c \ast f$. This operator has a second MUM point at infinity with the same instanton numbers. It can be
reduced to an operator of degree 4 with a single MUM-point over
$Q(\sqrt{?})$

373

New Number: 8.12 | AESZ: 175 | Superseeker: 17 1387 | Hash: f6db11b5e593983f455489d5bb1003c5

Degree: 8

\(\theta^4-x(10\theta^2+10\theta+3)(17\theta^2+17\theta+6)+3^{4} x^{2}\left(89\theta^4+452\theta^3+633\theta^2+362\theta+80\right)+2^{3} 3^{4} x^{3}\left(170\theta^4-1020\theta^3-3119\theta^2-2373\theta-648\right)-2^{4} 3^{8} x^{4}\left(97\theta^4+194\theta^3-238\theta^2-335\theta-114\right)+2^{6} 3^{8} x^{5}\left(170\theta^4+1700\theta^3+961\theta^2-125\theta-204\right)+2^{6} 3^{12} x^{6}\left(89\theta^4-96\theta^3-189\theta^2-96\theta-12\right)-2^{9} 3^{12} x^{7}(10\theta^2+10\theta+3)(17\theta^2+17\theta+6)+2^{12} 3^{16} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 18, 630, 29016, 1529766, ...
--> OEIS
Normalized instanton numbers (n₀=1): 17, -299/4, 1387, -47623/2, 500282, ... ; Common denominator:...

Discriminant

\((81z-1)(8z-1)(72z-1)(9z-1)(-1+648z^2)^2\)

Local exponents

\(-\frac{ 1}{ 36}\sqrt{ 2}\) \(0\) \(\frac{ 1}{ 81}\) \(\frac{ 1}{ 72}\) \(\frac{ 1}{ 36}\sqrt{ 2}\) \(\frac{ 1}{ 9}\) \(\frac{ 1}{ 8}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(3\) \(0\) \(1\) \(1\) \(3\) \(1\) \(1\) \(1\)
\(4\) \(0\) \(2\) \(2\) \(4\) \(2\) \(2\) \(1\)

Note:

Hadamard product $c \ast g$. This operator has a second MUM-point
at infinity with the same instanton numbers. It can be reduced to an operator of degree 4 with a single MUM-point defined over $Q(\sqrt{?})$.

374

New Number: 8.13 | AESZ: 163 | Superseeker: 12 3020/3 | Hash: e21fd830a9dca03305deb8363a26fcf2

Degree: 8

\(\theta^4-2^{2} 3 x\left((3\theta^2+3\theta+1)^2\right)+2^{4} 3^{2} x^{2}\left(21\theta^4+156\theta^3+219\theta^2+126\theta+29\right)+2^{7} 3^{4} x^{3}(3\theta^2+3\theta+1)(3\theta^2-21\theta-35)-2^{10} 3^{5} x^{4}\left(27\theta^4+54\theta^3-114\theta^2-141\theta-49\right)+2^{12} 3^{7} x^{5}(3\theta^2+3\theta+1)(3\theta^2+27\theta-11)+2^{14} 3^{8} x^{6}\left(21\theta^4-72\theta^3-123\theta^2-72\theta-13\right)-2^{17} 3^{10} x^{7}\left((3\theta^2+3\theta+1)^2\right)+2^{20} 3^{12} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 12, 180, 2352, 6084, ...
--> OEIS
Normalized instanton numbers (n₀=1): 12, -96, 3020/3, -71493/4, 319584, ... ; Common denominator:...

Discriminant

\((1728z^2-72z+1)(432z^2-36z+1)(-1+864z^2)^2\)

Local exponents

\(-\frac{ 1}{ 72}\sqrt{ 6}\) \(0\) \(\frac{ 1}{ 48}-\frac{ 1}{ 144}\sqrt{ 3}I\) \(\frac{ 1}{ 48}+\frac{ 1}{ 144}\sqrt{ 3}I\) \(\frac{ 1}{ 72}\sqrt{ 6}\) \(\frac{ 1}{ 24}-\frac{ 1}{ 72}\sqrt{ 3}I\) \(\frac{ 1}{ 24}+\frac{ 1}{ 72}\sqrt{ 3}I\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(3\) \(0\) \(1\) \(1\) \(3\) \(1\) \(1\) \(1\)
\(4\) \(0\) \(2\) \(2\) \(4\) \(2\) \(2\) \(1\)

Note:

Hadamard product $d \ast f$. This operator has a second MUM-point at infinity with the same instanton numbers. Itg can be reduced to an operator of degree 4 with a single MUM-point defined over $Q(\sqrt{?})$.

375

New Number: 8.14 | AESZ: 176 | Superseeker: 24 15448/3 | Hash: e2a40a57f7e88dba6655d936b4abe327

Degree: 8

\(\theta^4-2^{2} x(3\theta^2+3\theta+1)(17\theta^2+17\theta+6)+2^{5} x^{2}\left(325\theta^4+2164\theta^3+3053\theta^2+1778\theta+420\right)+2^{10} 3^{2} x^{3}\left(51\theta^4-306\theta^3-934\theta^2-717\theta-204\right)-2^{14} 3^{2} x^{4}\left(397\theta^4+794\theta^3-1454\theta^2-1851\theta-666\right)+2^{18} 3^{4} x^{5}\left(51\theta^4+510\theta^3+290\theta^2-29\theta-64\right)+2^{21} 3^{4} x^{6}\left(325\theta^4-864\theta^3-1489\theta^2-864\theta-144\right)-2^{26} 3^{6} x^{7}(3\theta^2+3\theta+1)(17\theta^2+17\theta+6)+2^{32} 3^{8} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 24, 840, 34944, 1618344, ...
--> OEIS
Normalized instanton numbers (n₀=1): 24, -509/2, 15448/3, -128530, 3746624, ... ; Common denominator:...

Discriminant

\((72z-1)(36z-1)(64z-1)(32z-1)(48z-1)^2(48z+1)^2\)

Local exponents

\(-\frac{ 1}{ 48}\) \(0\) \(\frac{ 1}{ 72}\) \(\frac{ 1}{ 64}\) \(\frac{ 1}{ 48}\) \(\frac{ 1}{ 36}\) \(\frac{ 1}{ 32}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(3\) \(0\) \(1\) \(1\) \(3\) \(1\) \(1\) \(1\)
\(4\) \(0\) \(2\) \(2\) \(4\) \(2\) \(2\) \(1\)

Note:

Hadamard product $d \ast g$. This operator has a second MUM-point at infinity with the same instanton numbers. It
can be reduced to an operator of degree 4 with a single MUM-point defined over $Q(\sqrt{?})$.

376

New Number: 8.15 | AESZ: 178 | Superseeker: 18 9799/3 | Hash: e748913f322a008ae5c350f96f1cd860

Degree: 8

\(\theta^4-3 x(3\theta^2+3\theta+1)(17\theta^2+17\theta+6)+3^{3} x^{2}\left(217\theta^4+1732\theta^3+2441\theta^2+1418\theta+336\right)+2^{3} 3^{6} x^{3}\left(51\theta^4-306\theta^3-934\theta^2-717\theta-204\right)-2^{4} 3^{8} x^{4}\left(289\theta^4+578\theta^3-1310\theta^2-1599\theta-570\right)+2^{6} 3^{11} x^{5}\left(51\theta^4+510\theta^3+290\theta^2-29\theta-64\right)+2^{6} 3^{13} x^{6}\left(217\theta^4-864\theta^3-1453\theta^2-864\theta-156\right)-2^{9} 3^{16} x^{7}(3\theta^2+3\theta+1)(17\theta^2+17\theta+6)+2^{12} 3^{20} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 18, 378, 6552, 21546, ...
--> OEIS
Normalized instanton numbers (n₀=1): 18, -423/2, 9799/3, -150003/2, 1914237, ... ; Common denominator:...

Discriminant

\((1728z^2-72z+1)(2187z^2-81z+1)(-1+1944z^2)^2\)

Local exponents

\(-\frac{ 1}{ 108}\sqrt{ 6}\) \(0\) \(\frac{ 1}{ 54}-\frac{ 1}{ 162}\sqrt{ 3}I\) \(\frac{ 1}{ 54}+\frac{ 1}{ 162}\sqrt{ 3}I\) \(\frac{ 1}{ 48}-\frac{ 1}{ 144}\sqrt{ 3}I\) \(\frac{ 1}{ 48}+\frac{ 1}{ 144}\sqrt{ 3}I\) \(\frac{ 1}{ 108}\sqrt{ 6}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(3\) \(0\) \(1\) \(1\) \(1\) \(1\) \(3\) \(1\)
\(4\) \(0\) \(2\) \(2\) \(2\) \(2\) \(4\) \(1\)

Note:

Hadamard product $d \ast g$. This operator has a second MUM-point at infinity. It can be reduced to an operator of degree 4 with a single MUM-point defined over $Q(\sqrt{?})$.

377

New Number: 8.16 | AESZ: 196 | Superseeker: 189/47 9277/47 | Hash: fdeee36c14d9c003b59c1738c024d479

Degree: 8

\(47^{2} \theta^4-47 x\left(2489\theta^4+4984\theta^3+4043\theta^2+1551\theta+235\right)-x^{2}\left(161022+701851\theta+1135848\theta^2+790072\theta^3+208867\theta^4\right)+x^{3}\left(38352+149319\theta+383912\theta^2+637644\theta^3+370857\theta^4\right)-x^{4}\left(1770676+5161283\theta+4424049\theta^2+511820\theta^3-291161\theta^4\right)+x^{5}\left(2151-260936\theta-750755\theta^2-749482\theta^3-406192\theta^4\right)+3^{3} x^{6}\left(5305\theta^4+90750\theta^3+152551\theta^2+91194\theta+17914\right)+2 3^{6} x^{7}\left(106\theta^4+230\theta^3+197\theta^2+82\theta+15\right)-2^{2} 3^{10} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 5, 93, 2507, 81229, ...
--> OEIS
Normalized instanton numbers (n₀=1): 189/47, 979/47, 9277/47, 124795/47, 2049020/47, ... ; Common denominator:...

Discriminant

\(-(-1+53z+90z^2-50z^3+81z^4)(-47-z+54z^2)^2\)

Local exponents

\(\frac{ 1}{ 108}-\frac{ 1}{ 108}\sqrt{ 10153}\) \(0\) \(\frac{ 1}{ 108}+\frac{ 1}{ 108}\sqrt{ 10153}\) \(#ND+#NDI\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(0\) \(1\) \(1\) \(1\)
\(3\) \(0\) \(3\) \(1\) \(1\)
\(4\) \(0\) \(4\) \(2\) \(1\)

Note:

The operator has a second MUM-point at infinity, corresponding to operator 8.17 .

378

New Number: 8.17 | AESZ: 200 | Superseeker: 19/2 -99607/18 | Hash: e970fa76e74543660fe271b31c8ad485

Degree: 8

\(2^{2} \theta^4-2 x\left(106\theta^4+194\theta^3+143\theta^2+46\theta+6\right)-3 x^{2}\left(5305\theta^4-69530\theta^3-87869\theta^2-37122\theta-6174\right)+3^{2} x^{3}\left(406192\theta^4+875286\theta^3+939461\theta^2+616896\theta+144378\right)-3^{6} x^{4}\left(291161\theta^4+1676464\theta^3-1141623\theta^2-986711\theta-230461\right)-3^{10} x^{5}\left(370857\theta^4+845784\theta^3+696122\theta^2+189001\theta+6158\right)+3^{14} x^{6}\left(208867\theta^4+45396\theta^3+18834\theta^2+35097\theta+13814\right)+3^{18} 47 x^{7}\left(2489\theta^4+4972\theta^3+4025\theta^2+1539\theta+232\right)-3^{22} 47^{2} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 3, -243, -15315, -55971, ...
--> OEIS
Normalized instanton numbers (n₀=1): 19/2, -5195/8, -99607/18, -217650, 23603349/2, ... ; Common denominator:...

Discriminant

\(-(531441z^4-347733z^3-7290z^2+50z-1)(-2+3z+11421z^2)^2\)

Local exponents

\(-\frac{ 1}{ 7614}-\frac{ 1}{ 7614}\sqrt{ 10153}\) \(0\) \(-\frac{ 1}{ 7614}+\frac{ 1}{ 7614}\sqrt{ 10153}\) \(#ND+#NDI\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(0\) \(1\) \(1\) \(1\)
\(3\) \(0\) \(3\) \(1\) \(1\)
\(4\) \(0\) \(4\) \(2\) \(1\)

Note:

This operator has a second MUM-point at infinity, corresponding to operator 8.16

379

New Number: 8.18 | AESZ: 197 | Superseeker: 3 1621/13 | Hash: 4cc8bdba73e5fa6cb4089fa5296429de

Degree: 8

\(13^{2} \theta^4-13^{2} x\left(41\theta^4+82\theta^3+67\theta^2+26\theta+4\right)-2^{3} 13 x^{2}\left(471\theta^4+1788\theta^3+2555\theta^2+1534\theta+338\right)+2^{6} 13 x^{3}\left(251\theta^4+1014\theta^3+1798\theta^2+1413\theta+405\right)+2^{9} x^{4}\left(749\theta^4+436\theta^3-4908\theta^2-6266\theta-2145\right)-2^{12} x^{5}\left(379\theta^4+1270\theta^3+967\theta^2-42\theta-178\right)-2^{15} x^{6}\left(9\theta^4-156\theta^3-273\theta^2-156\theta-28\right)+2^{18} x^{7}\left(13\theta^4+26\theta^3+20\theta^2+7\theta+1\right)-2^{21} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 4, 68, 1552, 43156, ...
--> OEIS
Normalized instanton numbers (n₀=1): 3, 226/13, 1621/13, 20666/13, 289056/13, ... ; Common denominator:...

Discriminant

\(-(z-1)(8z+1)(64z^2-48z+1)(-13+64z^2)^2\)

Local exponents

\(-\frac{ 1}{ 8}\sqrt{ 13}\) \(-\frac{ 1}{ 8}\) \(0\) \(\frac{ 3}{ 8}-\frac{ 1}{ 4}\sqrt{ 2}\) \(\frac{ 1}{ 8}\sqrt{ 13}\) \(\frac{ 3}{ 8}+\frac{ 1}{ 4}\sqrt{ 2}\) \(1\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(3\) \(1\) \(0\) \(1\) \(3\) \(1\) \(1\) \(1\)
\(4\) \(2\) \(0\) \(2\) \(4\) \(2\) \(2\) \(1\)

Note:

The operator has a second MUM-point at infinity, corresponding to operator 8.19.

380

New Number: 8.19 | AESZ: 201 | Superseeker: 32 7584 | Hash: d21570c07bca6887061716b2d727fa75

Degree: 8

\(\theta^4-2^{4} x\left(13\theta^4+26\theta^3+20\theta^2+7\theta+1\right)+2^{8} x^{2}\left(9\theta^4+192\theta^3+249\theta^2+114\theta+20\right)+2^{12} x^{3}\left(379\theta^4+246\theta^3-569\theta^2-318\theta-60\right)-2^{16} x^{4}\left(749\theta^4+2560\theta^3-1722\theta^2-1862\theta-474\right)-2^{20} 13 x^{5}\left(251\theta^4-10\theta^3+262\theta^2+145\theta+27\right)+2^{24} 13 x^{6}\left(471\theta^4+96\theta^3+17\theta^2+96\theta+42\right)+2^{28} 13^{2} x^{7}\left(41\theta^4+82\theta^3+67\theta^2+26\theta+4\right)-2^{35} 13^{2} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 16, 752, 49408, 3805456, ...
--> OEIS
Normalized instanton numbers (n₀=1): 32, -152, 7584, -160593, 7055200, ... ; Common denominator:...

Discriminant

\(-(128z-1)(16z+1)(256z^2-96z+1)(-1+3328z^2)^2\)

Local exponents

\(-\frac{ 1}{ 16}\) \(-\frac{ 1}{ 208}\sqrt{ 13}\) \(0\) \(\frac{ 1}{ 128}\) \(\frac{ 3}{ 16}-\frac{ 1}{ 8}\sqrt{ 2}\) \(\frac{ 1}{ 208}\sqrt{ 13}\) \(\frac{ 3}{ 16}+\frac{ 1}{ 8}\sqrt{ 2}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(1\) \(3\) \(0\) \(1\) \(1\) \(3\) \(1\) \(1\)
\(2\) \(4\) \(0\) \(2\) \(2\) \(4\) \(2\) \(1\)

Note:

This operator has a second MUM-point, corresponding to operator 8.18

381

New Number: 8.1 | AESZ: 102 | Superseeker: 8 1053 | Hash: e928905653beb9d844e6a942f50d94ac

Degree: 8

\(\theta^4-x(7\theta^2+7\theta+2)(11\theta^2+11\theta+3)-x^{2}\left(1049\theta^4+4100\theta^3+5689\theta^2+3178\theta+640\right)+2^{3} x^{3}\left(77\theta^4-462\theta^3-1420\theta^2-1053\theta-252\right)+2^{4} x^{4}\left(1041\theta^4+2082\theta^3-1406\theta^2-2447\theta-746\right)+2^{6} x^{5}\left(77\theta^4+770\theta^3+428\theta^2-93\theta-80\right)-2^{6} x^{6}\left(1049\theta^4+96\theta^3-317\theta^2+96\theta+100\right)-2^{9} x^{7}(7\theta^2+7\theta+2)(11\theta^2+11\theta+3)+2^{12} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 6, 190, 8232, 432846, ...
--> OEIS
Normalized instanton numbers (n₀=1): 8, 153/2, 1053, 49101/2, 670214, ... ; Common denominator:...

Discriminant

\((64z^2+88z-1)(z^2-11z-1)(-1+8z^2)^2\)

Local exponents

\(-\frac{ 11}{ 16}-\frac{ 5}{ 16}\sqrt{ 5}\) \(-\frac{ 1}{ 4}\sqrt{ 2}\) \(\frac{ 11}{ 2}-\frac{ 5}{ 2}\sqrt{ 5}\) \(0\) \(-\frac{ 11}{ 16}+\frac{ 5}{ 16}\sqrt{ 5}\) \(\frac{ 1}{ 4}\sqrt{ 2}\) \(\frac{ 11}{ 2}+\frac{ 5}{ 2}\sqrt{ 5}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\)
\(1\) \(3\) \(1\) \(0\) \(1\) \(3\) \(1\) \(1\)
\(2\) \(4\) \(2\) \(0\) \(2\) \(4\) \(2\) \(1\)

Note:

Hadamard product $a \ast b$. The operator has a second MUM-point at infinity with the same instanton numbers. In fact, there is a symmetry in the operator. It can be reduced to an operator with a single MUM point of degree 4, defined over $Q(\sqrt{2})$.

382

New Number: 8.20 | AESZ: 213 | Superseeker: 118/17 672 | Hash: d430b37f4ca641af0b82cbef83547c51

Degree: 8

\(17^{2} \theta^4-2 17 x\left(647\theta^4+1240\theta^3+977\theta^2+357\theta+51\right)-2^{2} x^{2}\left(14437\theta^4+89752\theta^3+147734\theta^2+92123\theta+20400\right)+2^{2} 3 x^{3}\left(21538\theta^4+25680\theta^3-41979\theta^2-56151\theta-17442\right)+2^{3} x^{4}\left(51920\theta^4+166384\theta^3-83149\theta^2-217017\theta-79362\right)-2^{4} 3 x^{5}\left(9360\theta^4-26784\theta^3-43813\theta^2-21965\theta-3496\right)-2^{5} 3 x^{6}\left(10160\theta^4-96\theta^3-10535\theta^2-5385\theta-438\right)-2^{8} 3^{2} x^{7}\left(288\theta^4+864\theta^3+1082\theta^2+641\theta+147\right)-2^{11} 3^{2} x^{8}(4\theta+3)(\theta+1)^2(4\theta+5)\)

Coefficients of the holomorphic solution: 1, 6, 162, 6252, 290610, ...
--> OEIS
Normalized instanton numbers (n₀=1): 118/17, 873/17, 672, 447987/34, 5358846/17, ... ; Common denominator:...

Discriminant

\(-(4z+1)(32z^3+40z^2+78z-1)(-17+18z+48z^2)^2\)

Local exponents

\(-\frac{ 3}{ 16}-\frac{ 1}{ 48}\sqrt{ 897}\) ≈\(-0.631368-1.433512I\) ≈\(-0.631368+1.433512I\) \(-\frac{ 1}{ 4}\) \(0\) ≈\(0.012736\) \(-\frac{ 3}{ 16}+\frac{ 1}{ 48}\sqrt{ 897}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(\frac{ 3}{ 4}\)
\(1\) \(1\) \(1\) \(1\) \(0\) \(1\) \(1\) \(1\)
\(3\) \(1\) \(1\) \(1\) \(0\) \(1\) \(3\) \(1\)
\(4\) \(2\) \(2\) \(2\) \(0\) \(2\) \(4\) \(\frac{ 5}{ 4}\)

383

New Number: 8.21 | AESZ: 251 | Superseeker: -9 -3145/3 | Hash: dd2b60d18804c72129ba319fc8b50023

Degree: 8

\(\theta^4-3 x\theta(-2-11\theta-18\theta^2+27\theta^3)-2 3^{2} x^{2}\left(39\theta^4+480\theta^3+474\theta^2+276\theta+64\right)+2^{3} 3^{4} x^{3}\left(348\theta^4+1152\theta^3+1759\theta^2+1110\theta+260\right)-2^{3} 3^{5} x^{4}\left(3420\theta^4+15912\theta^3+28437\theta^2+20544\theta+5296\right)+2^{4} 3^{7} x^{5}\left(1125\theta^4+12546\theta^3+31089\theta^2+26448\theta+7480\right)+2^{5} 3^{9} x^{6}\left(1395\theta^4+3240\theta^3-3378\theta^2-7146\theta-2696\right)-2^{7} 3^{11} x^{7}\left(351\theta^4+2646\theta^3+4767\theta^2+3309\theta+800\right)-2^{7} 3^{13} x^{8}(3\theta+2)(3\theta+4)(6\theta+5)(6\theta+7)\)

Coefficients of the holomorphic solution: 1, 0, 72, -1440, 48600, ...
--> OEIS
Normalized instanton numbers (n₀=1): -9, -27/4, -3145/3, -20907/4, -327348, ... ; Common denominator:...

Discriminant

\(-(54z+1)(27z-1)(432z^2-36z+1)(-1+36z+324z^2)^2\)

Local exponents

\(-\frac{ 1}{ 18}-\frac{ 1}{ 18}\sqrt{ 2}\) \(-\frac{ 1}{ 54}\) \(0\) \(-\frac{ 1}{ 18}+\frac{ 1}{ 18}\sqrt{ 2}\) \(\frac{ 1}{ 27}\) \(\frac{ 1}{ 24}-\frac{ 1}{ 72}\sqrt{ 3}I\) \(\frac{ 1}{ 24}+\frac{ 1}{ 72}\sqrt{ 3}I\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(\frac{ 2}{ 3}\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(\frac{ 5}{ 6}\)
\(3\) \(1\) \(0\) \(3\) \(1\) \(1\) \(1\) \(\frac{ 7}{ 6}\)
\(4\) \(2\) \(0\) \(4\) \(2\) \(2\) \(2\) \(\frac{ 4}{ 3}\)

Note:

This is operator "8.21" from ...

384

New Number: 8.22 | AESZ: 284 | Superseeker: 241/38 8729/19 | Hash: dbe506beab1f66a0b331f15c91b7fcde

Degree: 8

\(2^{2} 19^{2} \theta^4-2 19 x\left(3014\theta^4+5878\theta^3+4725\theta^2+1786\theta+266\right)+x^{2}\left(402002+1810054\theta+3057079\theta^2+2305502\theta^3+689717\theta^4\right)-x^{3}\left(1576582+6295992\theta+9142457\theta^2+5812350\theta^3+1438808\theta^4\right)+x^{4}\left(663471+3375833\theta+6297445\theta^2+5075392\theta^3+1395491\theta^4\right)+x^{5}\left(52928-604005\theta-2407768\theta^2-2657224\theta^3-834163\theta^4\right)-x^{6}\left(4832-148359\theta-572576\theta^2-692484\theta^3-277543\theta^4\right)-11 x^{7}\left(4625\theta^4+9100\theta^3+6395\theta^2+1845\theta+178\right)-11^{2} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 7, 163, 5767, 247651, ...
--> OEIS
Normalized instanton numbers (n₀=1): 241/38, 1353/38, 8729/19, 150334/19, 6399445/38, ... ; Common denominator:...

Discriminant

\(-(-1+78z-374z^2+425z^3+z^4)(38-25z+11z^2)^2\)

Local exponents

\(0\) \(\frac{ 25}{ 22}-\frac{ 1}{ 22}\sqrt{ 1047}I\) \(\frac{ 25}{ 22}+\frac{ 1}{ 22}\sqrt{ 1047}I\) \(#ND+#NDI\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(1\)
\(0\) \(1\) \(1\) \(1\) \(1\)
\(0\) \(3\) \(3\) \(1\) \(1\)
\(0\) \(4\) \(4\) \(2\) \(1\)

Note:

This operator has a second MUM-point at infinity, corresponding to operator 8.23

385

New Number: 8.23 | AESZ: 285 | Superseeker: -795/11 -1594688/11 | Hash: 009974f32940428eb2d2d31380b138a9

Degree: 8

\(11^{2} \theta^4+11 x\left(4625\theta^4+9400\theta^3+6845\theta^2+2145\theta+253\right)-x^{2}\left(4444+29513\theta+160382\theta^2+417688\theta^3+277543\theta^4\right)+x^{3}\left(834163\theta^4+679428\theta^3-558926\theta^2-423489\theta-72226\right)+x^{4}\left(94818+425155\theta+555785\theta^2-506572\theta^3-1395491\theta^4\right)+x^{5}\left(1438808\theta^4-57118\theta^3+338255\theta^2+307104\theta+49505\right)-x^{6}\left(33242+146466\theta+278875\theta^2+453366\theta^3+689717\theta^4\right)+2 19 x^{7}\left(3014\theta^4+6178\theta^3+5175\theta^2+2086\theta+341\right)-2^{2} 19^{2} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, -23, 3043, -620663, 154394851, ...
--> OEIS
Normalized instanton numbers (n₀=1): -795/11, 89027/44, -1594688/11, 166273857/11, -21441641455/11, ... ; Common denominator:...

Discriminant

\(-(-1-425z+374z^2-78z^3+z^4)(11-25z+38z^2)^2\)

Local exponents

\(0\) \(\frac{ 25}{ 76}-\frac{ 1}{ 76}\sqrt{ 1047}I\) \(\frac{ 25}{ 76}+\frac{ 1}{ 76}\sqrt{ 1047}I\) \(#ND+#NDI\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(1\)
\(0\) \(1\) \(1\) \(1\) \(1\)
\(0\) \(3\) \(3\) \(1\) \(1\)
\(0\) \(4\) \(4\) \(2\) \(1\)

Note:

This operator has a second MUM-point at infinity, corresponding to operator 8.22

386

New Number: 8.24 | AESZ: 286 | Superseeker: 3 437/3 | Hash: 94afcd38a40c3a3e54fc3c57b4b85459

Degree: 8

\(3^{2} \theta^4-3^{2} x\left(38\theta^4+82\theta^3+67\theta^2+26\theta+4\right)-3 x^{2}\left(2045\theta^4+5702\theta^3+7535\theta^2+4170\theta+852\right)+2^{3} 3 x^{3}\left(2208\theta^4+5925\theta^3+7925\theta^2+5607\theta+1512\right)+2^{3} x^{4}\left(60287\theta^4+56374\theta^3-215983\theta^2-268986\theta-85452\right)-2^{4} x^{5}\left(205651\theta^4+605608\theta^3+603579\theta^2+204622\theta+8104\right)-2^{7} x^{6}\left(51414\theta^4-273267\theta^3-502700\theta^2-305649\theta-63398\right)+2^{8} 37 x^{7}\left(7909\theta^4+18122\theta^3+17595\theta^2+8462\theta+1672\right)-2^{13} 37^{2} x^{8}(4\theta+3)(\theta+1)^2(4\theta+5)\)

Coefficients of the holomorphic solution: 1, 4, 72, 1696, 49960, ...
--> OEIS
Normalized instanton numbers (n₀=1): 3, 539/24, 437/3, 18531/8, 90274/3, ... ; Common denominator:...

Discriminant

\(-(-1+40z+504z^2-3088z^3+8192z^4)(-3-3z+148z^2)^2\)

Local exponents

\(\frac{ 3}{ 296}-\frac{ 1}{ 296}\sqrt{ 1785}\) ≈\(-0.070843\) \(0\) ≈\(0.020383\) \(\frac{ 3}{ 296}+\frac{ 1}{ 296}\sqrt{ 1785}\) ≈\(0.213707\) ≈\(0.213707\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(\frac{ 3}{ 4}\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(3\) \(1\) \(0\) \(1\) \(3\) \(1\) \(1\) \(1\)
\(4\) \(2\) \(0\) \(2\) \(4\) \(2\) \(2\) \(\frac{ 5}{ 4}\)

Note:

This is operator "8.24" from ...

387

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

Degree: 8

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

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 ...

388

New Number: 8.26 | AESZ: 301 | Superseeker: 193/11 48570/11 | Hash: a91db18876a9dfbf42b88f8d64c55d85

Degree: 8

\(11^{2} \theta^4-11 x\left(1517\theta^4+3136\theta^3+2393\theta^2+825\theta+110\right)-x^{2}\left(24266+106953\theta+202166\theta^2+207620\theta^3+90362\theta^4\right)-x^{3}\left(53130+217437\theta+415082\theta^2+507996\theta^3+245714\theta^4\right)-x^{4}\left(15226+183269\theta+564786\theta^2+785972\theta^3+407863\theta^4\right)-x^{5}\left(25160+279826\theta+728323\theta^2+790148\theta^3+434831\theta^4\right)-2^{3} x^{6}\left(36361\theta^4+70281\theta^3+73343\theta^2+37947\theta+7644\right)-2^{4} 5 x^{7}\left(1307\theta^4+3430\theta^3+3877\theta^2+2162\theta+488\right)-2^{9} 5^{2} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 10, 466, 32392, 2727826, ...
--> OEIS
Normalized instanton numbers (n₀=1): 193/11, 1973/11, 48570/11, 1689283/11, 72444183/11, ... ; Common denominator:...

Discriminant

\(-(-1+143z+32z^2)(z+1)^2(20z^2+17z+11)^2\)

Local exponents

\(-\frac{ 143}{ 64}-\frac{ 19}{ 64}\sqrt{ 57}\) \(-1\) \(-\frac{ 17}{ 40}-\frac{ 1}{ 40}\sqrt{ 591}I\) \(-\frac{ 17}{ 40}+\frac{ 1}{ 40}\sqrt{ 591}I\) \(0\) \(-\frac{ 143}{ 64}+\frac{ 19}{ 64}\sqrt{ 57}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(\frac{ 1}{ 2}\) \(1\) \(1\) \(0\) \(1\) \(1\)
\(1\) \(\frac{ 1}{ 2}\) \(3\) \(3\) \(0\) \(1\) \(1\)
\(2\) \(1\) \(4\) \(4\) \(0\) \(2\) \(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.27.

389

New Number: 8.27 | AESZ: 302 | Superseeker: 109/5 16777/5 | Hash: e18ddbe4d66a3648b349130bcf119dc7

Degree: 8

\(5^{2} \theta^4-5 x\left(1307\theta^4+1798\theta^3+1429\theta^2+530\theta+80\right)+2^{4} x^{2}\left(36361\theta^4+75163\theta^3+80666\theta^2+43340\theta+9120\right)-2^{6} x^{3}\left(434831\theta^4+949176\theta^3+966865\theta^2+545700\theta+118340\right)+2^{11} x^{4}\left(407863\theta^4+845480\theta^3+654048\theta^2+219839\theta+18634\right)-2^{16} x^{5}\left(245714\theta^4+474860\theta^3+365378\theta^2+71595\theta-11507\right)+2^{21} x^{6}\left(90362\theta^4+153828\theta^3+121478\theta^2+35967\theta+2221\right)-2^{26} 11 x^{7}\left(1517\theta^4+2932\theta^3+2087\theta^2+621\theta+59\right)-2^{31} 11^{2} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 16, 664, 41920, 3350776, ...
--> OEIS
Normalized instanton numbers (n₀=1): 109/5, 867/4, 16777/5, 662976/5, 26339071/5, ... ; Common denominator:...

Discriminant

\(-(-1+143z+32z^2)(32z-1)^2(2816z^2-136z+5)^2\)

Local exponents

\(-\frac{ 143}{ 64}-\frac{ 19}{ 64}\sqrt{ 57}\) \(0\) \(-\frac{ 143}{ 64}+\frac{ 19}{ 64}\sqrt{ 57}\) \(\frac{ 17}{ 704}-\frac{ 1}{ 704}\sqrt{ 591}I\) \(\frac{ 17}{ 704}+\frac{ 1}{ 704}\sqrt{ 591}I\) \(\frac{ 1}{ 32}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(0\) \(1\) \(1\) \(1\) \(\frac{ 1}{ 2}\) \(1\)
\(1\) \(0\) \(1\) \(3\) \(3\) \(\frac{ 1}{ 2}\) \(1\)
\(2\) \(0\) \(2\) \(4\) \(4\) \(1\) \(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.26

390

New Number: 8.28 | AESZ: 303 | Superseeker: 151/13 26293/13 | Hash: e081c85684dd16a72eeaf5a1b139b912

Degree: 8

\(13^{2} \theta^4-13 x\left(1505\theta^4+2746\theta^3+2127\theta^2+754\theta+104\right)+2^{2} x^{2}\left(22961\theta^4-2086\theta^3-55741\theta^2-41574\theta-9256\right)+2^{5} x^{3}\left(7524\theta^4+28098\theta^3+16131\theta^2+2691\theta-52\right)-2^{7} x^{4}\left(7241\theta^4+6214\theta^3+17522\theta^2+15423\theta+4146\right)-2^{8} x^{5}\left(6087\theta^4+1806\theta^3-3905\theta^2-3796\theta-1036\right)+2^{10} x^{6}\left(553\theta^4+4062\theta^3+4405\theta^2+1752\theta+220\right)+2^{14} x^{7}\left(82\theta^4+230\theta^3+275\theta^2+160\theta+37\right)+2^{18} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 8, 292, 15776, 1030036, ...
--> OEIS
Normalized instanton numbers (n₀=1): 151/13, 1436/13, 26293/13, 719465/13, 24184128/13, ... ; Common denominator:...

Discriminant

\((z-1)(64z^3+304z^2+108z-1)(-13+44z+64z^2)^2\)

Local exponents

≈\(-4.362346\) \(-\frac{ 11}{ 32}-\frac{ 1}{ 32}\sqrt{ 329}\) ≈\(-0.396684\) \(0\) ≈\(0.009029\) \(-\frac{ 11}{ 32}+\frac{ 1}{ 32}\sqrt{ 329}\) \(1\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\)
\(1\) \(3\) \(1\) \(0\) \(1\) \(3\) \(1\) \(1\)
\(2\) \(4\) \(2\) \(0\) \(2\) \(4\) \(2\) \(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.29

1-30  31-60  61-90  91-120  121-150  151-180
181-210  211-240  241-270  271-300  301-330  331-360
361-390  391-420  421-450  451-480  481-482