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

Summary

You searched for: degz=8

Your search produced 88 matches
1-30 31-60 61-88

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1

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 $\Q(\sqrt{?})$.

2

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
$Q(\sqrt{?})$

3

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 $Q(\sqrt{?})$.

4

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 $Q(\sqrt{?})$.

5

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 $Q(\sqrt{?})$.

6

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 $Q(\sqrt{?})$.

7

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 .

8

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

9

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.

10

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

11

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 $Q(\sqrt{2})$.

12

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

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

13

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 ...

14

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

15

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

16

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 ...

17

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 ...

18

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.

19

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

20

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

21

New Number: 8.29 | AESZ: 304 | Superseeker: -5 -641 | Hash: cf055a245b1537ed4f2609fa56cf67aa

Degree: 8

\(\theta^4+x\left(82\theta^4+98\theta^3+77\theta^2+28\theta+4\right)-x^{2}\left(636+2916\theta+4463\theta^2+1850\theta^3-553\theta^4\right)-2^{2} x^{3}\left(6087\theta^4+22542\theta^3+27199\theta^2+14916\theta+3136\right)-2^{5} x^{4}\left(7241\theta^4+22750\theta^3+42326\theta^2+29943\theta+7272\right)+2^{7} x^{5}\left(7524\theta^4+1998\theta^3-23019\theta^2-24627\theta-7186\right)+2^{8} x^{6}\left(22961\theta^4+93930\theta^3+88283\theta^2+28194\theta+1624\right)-2^{10} 13 x^{7}\left(1505\theta^4+3274\theta^3+2919\theta^2+1282\theta+236\right)+2^{14} 13^{2} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, -4, 112, -3712, 155536, ...
--> OEIS
Normalized instanton numbers (n₀=1): -5, 469/8, -641, 50173/4, -276231, ... ; Common denominator:...

Discriminant

\((16z-1)(64z^3-432z^2-76z-1)(-1-11z+52z^2)^2\)

Local exponents

≈\(-0.157556\) \(\frac{ 11}{ 104}-\frac{ 1}{ 104}\sqrt{ 329}\) ≈\(-0.014327\) \(0\) \(\frac{ 1}{ 16}\) \(\frac{ 11}{ 104}+\frac{ 1}{ 104}\sqrt{ 329}\) ≈\(6.921883\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\)
\(1\) \(3\) \(1\) \(0\) \(1\) \(3\) \(1\) \(1\)
\(2\) \(4\) \(2\) \(0\) \(2\) \(4\) \(2\) \(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.28

22

New Number: 8.2 | AESZ: 104 | Superseeker: 7 1271/3 | Hash: d6bd0d1524954c8ce0a6421d295e9795

Degree: 8

\(\theta^4-x(10\theta^2+10\theta+3)(7\theta^2+7\theta+2)-x^{2}\left(71\theta^4+1148\theta^3+1591\theta^2+886\theta+192\right)-2^{3} 3^{2} x^{3}\left(70\theta^4-420\theta^3-1289\theta^2-963\theta-240\right)-2^{4} 3^{2} x^{4}\left(143\theta^4+286\theta^3-1138\theta^2-1281\theta-414\right)+2^{6} 3^{4} x^{5}\left(70\theta^4+700\theta^3+391\theta^2-75\theta-76\right)-2^{6} 3^{4} x^{6}\left(71\theta^4-864\theta^3-1427\theta^2-864\theta-180\right)+2^{9} 3^{6} x^{7}(10\theta^2+10\theta+3)(7\theta^2+7\theta+2)+2^{12} 3^{8} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 6, 150, 5208, 221094, ...
--> OEIS
Normalized instanton numbers (n₀=1): 7, 93/2, 1271/3, 18507/2, 190710, ... ; Common denominator:...

Discriminant

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

Local exponents

\(-1\) \(-\frac{ 1}{ 9}\) \(0-\frac{ 1}{ 12}\sqrt{ 2}I\) \(0\) \(0+\frac{ 1}{ 12}\sqrt{ 2}I\) \(\frac{ 1}{ 72}\) \(\frac{ 1}{ 8}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\)
\(1\) \(1\) \(3\) \(0\) \(3\) \(1\) \(1\) \(1\)
\(2\) \(2\) \(4\) \(0\) \(4\) \(2\) \(2\) \(1\)

Note:

Hadamard product $a \ast c$. This operator has a second MUM-point at infinity with the same instanton point.
It is reducible to an operator of degree 4 with a single MUM-point defined over $Q(\sqrt{-2})$

23

New Number: 8.30 | AESZ: 314 | Superseeker: 229/4 297111/4 | Hash: 893692ba7eb3effcbc0c3b48d405456a

Degree: 8

\(2^{4} \theta^4-2^{2} x\left(1282\theta^4+2618\theta^3+1909\theta^2+600\theta+72\right)-3^{2} x^{2}\left(9503\theta^4+26810\theta^3+31755\theta^2+15944\theta+2936\right)+3^{4} x^{3}\left(15627\theta^4-18288\theta^3-91412\theta^2-53256\theta-9688\right)+2 3^{6} x^{4}\left(15106\theta^4+20300\theta^3-20421\theta^2-23443\theta-5907\right)-2^{2} 3^{8} x^{5}\left(2072\theta^4-18256\theta^3-2563\theta^2+4626\theta+1495\right)-2^{2} 3^{10} x^{6}\left(6204\theta^4+360\theta^3-281\theta^2+1017\theta+434\right)-2^{5} 3^{12} x^{7}(2\theta+1)(100\theta^3+162\theta^2+95\theta+21)+2^{8} 3^{14} x^{8}(2\theta+1)(\theta+1)^2(2\theta+3)\)

Coefficients of the holomorphic solution: 1, 18, 1926, 310860, 61060230, ...
--> OEIS
Normalized instanton numbers (n₀=1): 229/4, 1293, 297111/4, 6150238, 2540085295/4, ... ; Common denominator:...

Discriminant

\((z-1)(11664z^3+3888z^2+324z-1)(-4-9z+648z^2)^2\)

Local exponents

≈\(-0.168156-0.022431I\) ≈\(-0.168156+0.022431I\) \(\frac{ 1}{ 144}-\frac{ 1}{ 144}\sqrt{ 129}\) \(0\) \(\frac{ 1}{ 18}2^(\frac{ 1}{ 3})+\frac{ 1}{ 36}2^(\frac{ 2}{ 3})-\frac{ 1}{ 9}\) \(\frac{ 1}{ 144}+\frac{ 1}{ 144}\sqrt{ 129}\) \(1\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(\frac{ 1}{ 2}\)
\(1\) \(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\)
\(1\) \(1\) \(3\) \(0\) \(1\) \(3\) \(1\) \(1\)
\(2\) \(2\) \(4\) \(0\) \(2\) \(4\) \(2\) \(\frac{ 3}{ 2}\)

Note:

This is operator "8.30" from ...

24

New Number: 8.31 | AESZ: 315 | Superseeker: 38 26135 | Hash: 44be55b95bb1c725c5aaa2c9a6635e89

Degree: 8

\(5^{2} \theta^4-5^{2} x\left(239\theta^4+496\theta^3+368\theta^2+120\theta+15\right)-2 3 5 x^{2}\left(1727\theta^4+3206\theta^3+2341\theta^2+1090\theta+245\right)-3^{2} 5 x^{3}\left(1519\theta^4+7338\theta^3+14271\theta^2+8340\theta+1690\right)+3^{3} x^{4}\left(10358\theta^4-16622\theta^3-49763\theta^2-37900\theta-10210\right)+3^{4} 5 x^{5}\left(922\theta^4+3526\theta^3-1357\theta^2-3028\theta-1031\right)-3^{5} x^{6}\left(1219\theta^4-6030\theta^3-6441\theta^2-1740\theta+160\right)-2^{2} 3^{6} x^{7}\left(162\theta^4+234\theta^3+65\theta^2-52\theta-25\right)-2^{4} 3^{8} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 15, 1179, 140505, 20362059, ...
--> OEIS
Normalized instanton numbers (n₀=1): 38, 3068/5, 26135, 7871998/5, 117518569, ... ; Common denominator:...

Discriminant

\(-(z+1)(81z^3+351z^2+246z-1)(-5-15z+36z^2)^2\)

Local exponents

≈\(-3.452681\) \(-1\) ≈\(-0.884694\) \(\frac{ 5}{ 24}-\frac{ 1}{ 24}\sqrt{ 105}\) \(0\) ≈\(0.004042\) \(\frac{ 5}{ 24}+\frac{ 1}{ 24}\sqrt{ 105}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(1\) \(1\) \(0\) \(1\) \(1\) \(1\)
\(1\) \(1\) \(1\) \(3\) \(0\) \(1\) \(3\) \(1\)
\(2\) \(2\) \(2\) \(4\) \(0\) \(2\) \(4\) \(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.32

25

New Number: 8.32 | AESZ: 317 | Superseeker: 69/4 14365/12 | Hash: cda8cce31025f51636125bea67a820d1

Degree: 8

\(2^{4} \theta^4-2^{2} 3 x\left(162\theta^4+414\theta^3+335\theta^2+128\theta+20\right)+3^{3} x^{2}\left(1219\theta^4+10906\theta^3+18963\theta^2+11824\theta+2708\right)+3^{5} 5 x^{3}\left(922\theta^4+162\theta^3-6403\theta^2-6576\theta-1964\right)-3^{7} x^{4}\left(10358\theta^4+58054\theta^3+62251\theta^2+29672\theta+4907\right)-3^{9} 5 x^{5}\left(1519\theta^4-1262\theta^3+1371\theta^2+4264\theta+1802\right)+2 3^{11} 5 x^{6}\left(1727\theta^4+3702\theta^3+3085\theta^2+882\theta+17\right)-3^{13} 5^{2} x^{7}\left(239\theta^4+460\theta^3+314\theta^2+84\theta+6\right)-3^{16} 5^{2} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 15, 459, 19545, 1019259, ...
--> OEIS
Normalized instanton numbers (n₀=1): 69/4, -30, 14365/12, 3015/2, 1376205/4, ... ; Common denominator:...

Discriminant

\(-(27z-1)(243z^3+2214z^2-117z+1)(-4-45z+405z^2)^2\)

Local exponents

≈\(-9.163702\) \(\frac{ 1}{ 18}-\frac{ 1}{ 90}\sqrt{ 105}\) \(0\) ≈\(0.010727\) \(\frac{ 1}{ 27}\) ≈\(0.041864\) \(\frac{ 1}{ 18}+\frac{ 1}{ 90}\sqrt{ 105}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(1\) \(3\) \(0\) \(1\) \(1\) \(1\) \(3\) \(1\)
\(2\) \(4\) \(0\) \(2\) \(2\) \(2\) \(4\) \(1\)

Note:

This operator has a second MUM-point at infininty corresponding to operator 8.31

26

New Number: 8.33 | AESZ: 322 | Superseeker: 4/3 95/3 | Hash: a19da26bf1a7748e3b7e6151e803da30

Degree: 8

\(3^{2} \theta^4+3 x\left(5\theta^4-122\theta^3-100\theta^2-39\theta-6\right)-x^{2}\left(5052+23736\theta+41729\theta^2+32600\theta^3+8603\theta^4\right)-2^{2} x^{3}\left(33304\theta^4+108297\theta^3+122347\theta^2+61470\theta+11712\right)-2^{2} x^{4}\left(180401\theta^4+547606\theta^3+638125\theta^2+339248\theta+69036\right)-2^{4} x^{5}\left(94934\theta^4+298745\theta^3+355667\theta^2+189660\theta+38224\right)-2^{4} x^{6}\left(73291\theta^4+204216\theta^3+190453\theta^2+68916\theta+6964\right)-2^{7} 3 x^{7}\left(811\theta^4+1886\theta^3+1804\theta^2+861\theta+174\right)-2^{10} 3^{2} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 2, 46, 632, 16846, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4/3, 18, 95/3, 14575/12, 18158/3, ... ; Common denominator:...

Discriminant

\(-(-1+13z+827z^2+1928z^3+64z^4)(3+22z+12z^2)^2\)

Local exponents

\(-\frac{ 11}{ 12}-\frac{ 1}{ 12}\sqrt{ 85}\) \(-\frac{ 11}{ 12}+\frac{ 1}{ 12}\sqrt{ 85}\) \(0\) \(#ND+#NDI\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(0\) \(1\) \(1\)
\(3\) \(3\) \(0\) \(1\) \(1\)
\(4\) \(4\) \(0\) \(2\) \(1\)

Note:

This operator has a second MUM-point at infininty corresponding to operator 8.34

27

New Number: 8.34 | AESZ: 323 | Superseeker: 100/3 73111/3 | Hash: 77c03b04c3a10350b5b0ccd2d204b18f

Degree: 8

\(3^{2} \theta^4-3 x\left(811\theta^4+1358\theta^3+1012\theta^2+333\theta+42\right)-x^{2}\left(2424+7494\theta-17551\theta^2-88948\theta^3-73291\theta^4\right)-2^{3} x^{3}\left(94934\theta^4+80991\theta^3+29036\theta^2+5175\theta+420\right)+2^{4} x^{4}\left(180401\theta^4+173998\theta^3+77713\theta^2+15788\theta+708\right)-2^{7} x^{5}\left(33304\theta^4+24919\theta^3-2720\theta^2-8451\theta-2404\right)+2^{8} x^{6}\left(8603\theta^4+1812\theta^3-4453\theta^2-3666\theta-952\right)+2^{11} 3 x^{7}\left(5\theta^4+142\theta^3+296\theta^2+225\theta+60\right)-2^{14} 3^{2} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, 14, 1054, 120776, 16816846, ...
--> OEIS
Normalized instanton numbers (n₀=1): 100/3, 1880/3, 73111/3, 4310384/3, 314245046/3, ... ; Common denominator:...

Discriminant

\(-(-1+241z-827z^2+104z^3+64z^4)(3-44z+48z^2)^2\)

Local exponents

\(0\) \(\frac{ 11}{ 24}-\frac{ 1}{ 24}\sqrt{ 85}\) \(\frac{ 11}{ 24}+\frac{ 1}{ 24}\sqrt{ 85}\) \(#ND+#NDI\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(1\)
\(0\) \(1\) \(1\) \(1\) \(1\)
\(0\) \(3\) \(3\) \(1\) \(1\)
\(0\) \(4\) \(4\) \(2\) \(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.33

28

New Number: 8.35 | AESZ: 326 | Superseeker: 11/13 385/39 | Hash: 946b91838924db64fe0ebdf0d473e621

Degree: 8

\(13^{2} \theta^4-13 x\theta(56\theta^3+178\theta^2+115\theta+26)-x^{2}\left(28466\theta^4+109442\theta^3+165603\theta^2+117338\theta+32448\right)-x^{3}\left(233114\theta^4+1257906\theta^3+2622815\theta^2+2467842\theta+872352\right)-x^{4}\left(989585\theta^4+6852298\theta^3+17737939\theta^2+19969754\theta+8108448\right)-x^{5}(\theta+1)(2458967\theta^3+18007287\theta^2+44047582\theta+35386584)-3^{2} x^{6}(\theta+1)(\theta+2)(393163\theta^2+2539029\theta+4164444)-3^{3} 11 x^{7}(\theta+3)(\theta+2)(\theta+1)(8683\theta+34604)-3^{3} 11^{2} 13 17 x^{8}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Coefficients of the holomorphic solution: 1, 0, 12, 96, 1116, ...
--> OEIS
Normalized instanton numbers (n₀=1): 11/13, 30/13, 385/39, 672/13, 4437/13, ... ; Common denominator:...

Discriminant

\(-(3z+1)(13z^2+5z+1)(153z^3+75z^2+14z-1)(13+11z)^2\)

Local exponents

\(-\frac{ 13}{ 11}\) \(-\frac{ 1}{ 3}\) ≈\(-0.272124-0.216493I\) ≈\(-0.272124+0.216493I\) \(-\frac{ 5}{ 26}-\frac{ 3}{ 26}\sqrt{ 3}I\) \(-\frac{ 5}{ 26}+\frac{ 3}{ 26}\sqrt{ 3}I\) \(0\) ≈\(0.054052\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(0\) \(1\) \(2\)
\(3\) \(1\) \(1\) \(1\) \(1\) \(1\) \(0\) \(1\) \(3\)
\(4\) \(2\) \(2\) \(2\) \(2\) \(2\) \(0\) \(2\) \(4\)

Note:

This opeerator is reducible to 6.25

29

New Number: 8.36 | AESZ: 327 | Superseeker: 24/29 284/29 | Hash: 586c1906112cbba9b2d54c57ce2add99

Degree: 8

\(29^{2} \theta^4+2 29 x\theta(24\theta^3-198\theta^2-128\theta-29)-2^{2} x^{2}\left(44284\theta^4+172954\theta^3+248589\theta^2+172057\theta+47096\right)-2^{2} x^{3}\left(525708\theta^4+2414772\theta^3+4447643\theta^2+3839049\theta+1275594\right)-2^{3} x^{4}\left(1415624\theta^4+7911004\theta^3+17395449\theta^2+17396359\theta+6496262\right)-2^{4} x^{5}(\theta+1)(2152040\theta^3+12186636\theta^2+24179373\theta+16560506)-2^{5} x^{6}(\theta+1)(\theta+2)(1912256\theta^2+9108540\theta+11349571)-2^{8} 41 x^{7}(\theta+3)(\theta+2)(\theta+1)(5671\theta+16301)-2^{8} 3 19 41^{2} x^{8}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Coefficients of the holomorphic solution: 1, 0, 14, 96, 1266, ...
--> OEIS
Normalized instanton numbers (n₀=1): 24/29, 72/29, 284/29, 1616/29, 10632/29, ... ; Common denominator:...

Discriminant

\(-(6z+1)(152z^3+84z^2+14z-1)(2z+1)^2(82z+29)^2\)

Local exponents

\(-\frac{ 1}{ 2}\) \(-\frac{ 29}{ 82}\) ≈\(-0.302804-0.180271I\) ≈\(-0.302804+0.180271I\) \(-\frac{ 1}{ 6}\) \(0\) ≈\(0.052976\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(\frac{ 1}{ 2}\) \(1\) \(1\) \(1\) \(1\) \(0\) \(1\) \(2\)
\(\frac{ 1}{ 2}\) \(3\) \(1\) \(1\) \(1\) \(0\) \(1\) \(3\)
\(1\) \(4\) \(2\) \(2\) \(2\) \(0\) \(2\) \(4\)

Note:

This operator is reducible to operator 6.23

30

New Number: 8.37 | AESZ: 345 | Superseeker: -12/11 357/11 | Hash: 60f282ab4e1936cd96eb5ba12983db2d

Degree: 8

\(11^{2} \theta^4+3 11 x\left(113\theta^4+184\theta^3+158\theta^2+66\theta+11\right)+2 x^{2}\left(28397\theta^4+95138\theta^3+128420\theta^2+77715\theta+17622\right)-3 x^{3}\left(3165\theta^4+180822\theta^3+560611\theta^2+539022\theta+167508\right)-3 x^{4}\left(233330\theta^4+1052614\theta^3+1424797\theta^2+774518\theta+145896\right)-3^{2} x^{5}\left(12866\theta^4-98902\theta^3-52127\theta^2+102028\theta+63723\right)+3^{2} x^{6}\left(183763\theta^4+473778\theta^3+427847\theta^2+147060\theta+11268\right)-2^{3} 3^{3} x^{7}\left(5006\theta^4+13414\theta^3+14935\theta^2+8228\theta+1869\right)+2^{6} 3^{7} x^{8}\left((\theta+1)^4\right)\)

Coefficients of the holomorphic solution: 1, -3, 9, 141, -3879, ...
--> OEIS
Normalized instanton numbers (n₀=1): -12/11, -28/11, 357/11, -1172/11, -5250/11, ... ; Common denominator:...

Discriminant

\((3z-1)(81z^3-457z^2-30z-1)(-11-21z+24z^2)^2\)

Local exponents

\(\frac{ 7}{ 16}-\frac{ 1}{ 48}\sqrt{ 1497}\) ≈\(-0.032637-0.033136I\) ≈\(-0.032637+0.033136I\) \(0\) \(\frac{ 1}{ 3}\) \(\frac{ 7}{ 16}+\frac{ 1}{ 48}\sqrt{ 1497}\) ≈\(5.707249\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(1\) \(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\)
\(3\) \(1\) \(1\) \(0\) \(1\) \(3\) \(1\) \(1\)
\(4\) \(2\) \(2\) \(0\) \(2\) \(4\) \(2\) \(1\)

Note:

This operator has a second MUM point at infinity corresponding to operator 8.38.

1-30 31-60 61-88