Summary

You searched for: inst=18

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1

New Number: 2.21 |  AESZ: 134  |  Superseeker: 18 -5177  |  Hash: cc6d92c4b8a8dadb92b447c54e3a2a2f  

Degree: 2

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

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Coefficients of the holomorphic solution: 1, 18, 810, 35280, 311850, ...
--> OEIS
Normalized instanton numbers (n0=1): 18, -207/2, -5177, -155979, -923301, ... ; Common denominator:...

Discriminant

\(1-243z+19683z^2\)

Local exponents

\(0\)\(\frac{ 1}{ 162}-\frac{ 1}{ 486}\sqrt{ 3}I\)\(\frac{ 1}{ 162}+\frac{ 1}{ 486}\sqrt{ 3}I\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 3}\)
\(0\)\(1\)\(1\)\(\frac{ 2}{ 3}\)
\(0\)\(1\)\(1\)\(\frac{ 4}{ 3}\)
\(0\)\(2\)\(2\)\(\frac{ 5}{ 3}\)

Note:

Hadamard product $B\ast f$

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2

New Number: 5.38 |  AESZ: 223  |  Superseeker: 18 64744/3  |  Hash: e3ab25cffe4a0968b175bd9e98c96427  

Degree: 5

\(\theta^4+2 3 x\theta(48\theta^3-12\theta^2-7\theta-1)+2^{2} 3^{3} x^{2}\left(392\theta^4+488\theta^3+775\theta^2+376\theta+64\right)+2^{4} 3^{5} x^{3}\left(1184\theta^4+3288\theta^3+3512\theta^2+1635\theta+278\right)+2^{6} 3^{8} x^{4}(169\theta^2+361\theta+238)(2\theta+1)^2+2^{11} 3^{11} x^{5}(2\theta+1)^2(2\theta+3)^2\)

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Coefficients of the holomorphic solution: 1, 0, -432, 7200, 1587600, ...
--> OEIS
Normalized instanton numbers (n0=1): 18, -873, 64744/3, -229968, -1628892, ... ; Common denominator:...

Discriminant

\((36z+1)(13824z^2+36z+1)(1+108z)^2\)

Local exponents

\(-\frac{ 1}{ 36}\)\(-\frac{ 1}{ 108}\)\(-\frac{ 1}{ 768}-\frac{ 5}{ 2304}\sqrt{ 15}I\)\(-\frac{ 1}{ 768}+\frac{ 5}{ 2304}\sqrt{ 15}I\)\(0\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(1\)\(1\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(3\)\(1\)\(1\)\(0\)\(\frac{ 3}{ 2}\)
\(2\)\(4\)\(2\)\(2\)\(0\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.38" from ...

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3

New Number: 6.7 |  AESZ:  |  Superseeker: -9 217/3  |  Hash: 9492f991c909a6774f5546668ff53b6a  

Degree: 6

\(\theta^4-3 x\left(42\theta^4+84\theta^3+77\theta^2+35\theta+6\right)+3^{3} x^{2}\left(291\theta^4+1164\theta^3+1747\theta^2+1166\theta+264\right)-2^{2} 3^{5} x^{3}\left(360\theta^4+2160\theta^3+4553\theta^2+3939\theta+1035\right)+2^{3} 3^{8} x^{4}\left(204\theta^4+1632\theta^3+4449\theta^2+4740\theta+1400\right)-2^{4} 3^{11} x^{5}(16\theta^2+80\theta+35)(2\theta+5)^2+2^{4} 3^{14} x^{6}(2\theta+11)(2\theta+7)(2\theta+5)(2\theta+1)\)

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Coefficients of the holomorphic solution: 1, 18, 378, 8820, 266490, ...
--> OEIS
Normalized instanton numbers (n0=1): -9, 18, 217/3, -9, -146079, ... ; Common denominator:...

Discriminant

\((27z-1)(34992z^3-1944z^2+27z-1)(-1+36z)^2\)

Local exponents

\(0\) ≈\(0.002095-0.023494I\) ≈\(0.002095+0.023494I\)\(\frac{ 1}{ 36}\)\(\frac{ 1}{ 27}\) ≈\(0.051365\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(\frac{ 5}{ 2}\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(\frac{ 7}{ 2}\)
\(0\)\(2\)\(2\)\(1\)\(2\)\(2\)\(\frac{ 11}{ 2}\)

Note:

This is operator "6.7" from ...

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4

New Number: 7.17 |  AESZ:  |  Superseeker: 0 18  |  Hash: c248fd7c807d0aae71ef687a9ee40c80  

Degree: 7

\(\theta^4+3 x\left(87\theta^4+84\theta^3+86\theta^2+44\theta+9\right)+2 3^{3} x^{2}\left(539\theta^4+1076\theta^3+1366\theta^2+880\theta+233\right)+2 3^{5} x^{3}\left(3699\theta^4+11424\theta^3+17579\theta^2+13389\theta+4088\right)+3^{7} x^{4}\left(30367\theta^4+128696\theta^3+235722\theta^2+205070\theta+69226\right)+3^{9} x^{5}\left(74547\theta^4+405660\theta^3+871096\theta^2+848930\theta+310507\right)+2 3^{11} 5 x^{6}(2\theta+3)(5066\theta^3+26325\theta^2+44815\theta+23766)+2^{2} 3^{14} 5^{2} 7^{2} x^{7}(\theta+1)(2\theta+5)(2\theta+3)(\theta+3)\)

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Coefficients of the holomorphic solution: 1, -27, 783, -23481, 717903, ...
--> OEIS
Normalized instanton numbers (n0=1): 0, -27/2, 18, -999/2, 1566, ... ; Common denominator:...

Discriminant

\((27z+1)(1323z^2+72z+1)(36z+1)^2(45z+1)^2\)

Local exponents

\(-\frac{ 1}{ 27}\)\(-\frac{ 1}{ 36}\)\(-\frac{ 4}{ 147}-\frac{ 1}{ 441}\sqrt{ 3}I\)\(-\frac{ 4}{ 147}+\frac{ 1}{ 441}\sqrt{ 3}I\)\(-\frac{ 1}{ 45}\)\(0\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)\(0\)\(\frac{ 3}{ 2}\)
\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(3\)\(0\)\(\frac{ 5}{ 2}\)
\(2\)\(1\)\(2\)\(2\)\(4\)\(0\)\(3\)

Note:

This is operator "7.17" from ...

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5

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

Degree: 8

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

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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 $Q(\sqrt{?})$.

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6

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

Degree: 8

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

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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

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