Summary

You searched for: inst=16

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1

New Number: 2.13 |  AESZ: 36  |  Superseeker: 16 1232  |  Hash: dea6fdf568a5907a24ba30fef2caf124  

Degree: 2

\(\theta^4-2^{4} x(2\theta+1)^2(3\theta^2+3\theta+1)+2^{9} x^{2}(2\theta+1)^2(2\theta+3)^2\)

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Coefficients of the holomorphic solution: 1, 16, 720, 44800, 3312400, ...
--> OEIS
Normalized instanton numbers (n0=1): 16, 42, 1232, 32159, 990128, ... ; Common denominator:...

Discriminant

\((128z-1)(64z-1)\)

Local exponents

\(0\)\(\frac{ 1}{ 128}\)\(\frac{ 1}{ 64}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(\frac{ 3}{ 2}\)
\(0\)\(2\)\(2\)\(\frac{ 3}{ 2}\)

Note:

A*d

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2

New Number: 2.9 |  AESZ: 58  |  Superseeker: 16 11056/3  |  Hash: 1ca6d3d1c4514db0651efce420265f5a  

Degree: 2

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

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Coefficients of the holomorphic solution: 1, 12, 540, 37200, 3131100, ...
--> OEIS
Normalized instanton numbers (n0=1): 16, 142, 11056/3, 121470, 4971792, ... ; Common denominator:...

Discriminant

\((144z-1)(16z-1)\)

Local exponents

\(0\)\(\frac{ 1}{ 144}\)\(\frac{ 1}{ 16}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(\frac{ 3}{ 2}\)
\(0\)\(2\)\(2\)\(\frac{ 3}{ 2}\)

Note:

Hadamard product A*c

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3

New Number: 3.34 |  AESZ:  |  Superseeker: 16 1744  |  Hash: 931a876bfe4d4aa192c6e18e74047640  

Degree: 3

\(\theta^4-2^{4} x\left(25\theta^4+50\theta^3+43\theta^2+18\theta+3\right)+2^{11} x^{2}(26\theta^2+52\theta+21)(\theta+1)^2-2^{16} 3^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)\)

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Coefficients of the holomorphic solution: 1, 48, 3984, 387840, 40818960, ...
--> OEIS
Normalized instanton numbers (n0=1): 16, -110, 1744, -29526, 644016, ... ; Common denominator:...

Discriminant

\(-(144z-1)(-1+128z)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 144}\)\(\frac{ 1}{ 128}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(2\)
\(0\)\(2\)\(1\)\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to AESZ 107 $=d \ast d$

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4

New Number: 8.41 |  AESZ:  |  Superseeker: 16 7568/3  |  Hash: 051d7068f49d14c45c3c3369d63d56b5  

Degree: 8

\(3^{2} \theta^4-2^{2} 3^{2} x\left(23\theta^4+58\theta^3+44\theta^2+15\theta+2\right)-2^{5} 3 x^{2}\left(254\theta^4+662\theta^3+623\theta^2+309\theta+66\right)-2^{8} 3 x^{3}\left(569\theta^4+1092\theta^3+602\theta^2+285\theta+78\right)-2^{11} x^{4}\left(2266\theta^4+4076\theta^3+2167\theta^2+537\theta+18\right)-2^{16} x^{5}\left(519\theta^4+798\theta^3+821\theta^2+391\theta+62\right)-2^{19} x^{6}\left(305\theta^4+558\theta^3+625\theta^2+360\theta+82\right)-2^{24} x^{7}\left(26\theta^4+70\theta^3+83\theta^2+48\theta+11\right)-2^{29} x^{8}\left((\theta+1)^4\right)\)

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Coefficients of the holomorphic solution: 1, 8, 328, 18944, 1324456, ...
--> OEIS
Normalized instanton numbers (n0=1): 16, 751/6, 7568/3, 229516/3, 8099456/3, ... ; Common denominator:...

Discriminant

\(-(4z+1)(2048z^3+768z^2+112z-1)(3+24z+256z^2)^2\)

Local exponents

\(-\frac{ 1}{ 4}\) ≈\(-0.191715-0.145483I\) ≈\(-0.191715+0.145483I\)\(-\frac{ 3}{ 64}-\frac{ 1}{ 64}\sqrt{ 39}I\)\(-\frac{ 3}{ 64}+\frac{ 1}{ 64}\sqrt{ 39}I\)\(0\) ≈\(0.00843\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)
\(1\)\(1\)\(1\)\(3\)\(3\)\(0\)\(1\)\(1\)
\(2\)\(2\)\(2\)\(4\)\(4\)\(0\)\(2\)\(1\)

Note:

This is operator "8.41" from ...

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5

New Number: 8.9 |  AESZ: 174  |  Superseeker: 16 -13  |  Hash: 3f987b46d9ebf201eeead1a885b78e66  

Degree: 8

\(\theta^4-x(11\theta^2+11\theta+3)(17\theta^2+17\theta+6)+x^{2}\left(8711\theta^4+33980\theta^3+47095\theta^2+26230\theta+5232\right)-2^{3} 3^{2} x^{3}\left(187\theta^4-1122\theta^3-3436\theta^2-2595\theta-684\right)+2^{4} 3^{2} x^{4}\left(8639\theta^4+17278\theta^3-11650\theta^2-20289\theta-6102\right)+2^{6} 3^{4} x^{5}\left(187\theta^4+1870\theta^3+1052\theta^2-163\theta-216\right)+2^{6} 3^{4} x^{6}\left(8711\theta^4+864\theta^3-2579\theta^2+864\theta+828\right)+2^{9} 3^{6} x^{7}(11\theta^2+11\theta+3)(17\theta^2+17\theta+6)+2^{12} 3^{8} x^{8}\left((\theta+1)^4\right)\)

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Coefficients of the holomorphic solution: 1, 18, 798, 45864, 2994894, ...
--> OEIS
Normalized instanton numbers (n0=1): 16, 7/2, -13, 11663/2, -26414, ... ; Common denominator:...

Discriminant

\((81z^2+99z-1)(64z^2+88z-1)(1+72z^2)^2\)

Local exponents

\(-\frac{ 11}{ 16}-\frac{ 5}{ 16}\sqrt{ 5}\)\(-\frac{ 11}{ 18}-\frac{ 5}{ 18}\sqrt{ 5}\)\(0-\frac{ 1}{ 12}\sqrt{ 2}I\)\(0\)\(0+\frac{ 1}{ 12}\sqrt{ 2}I\)\(-\frac{ 11}{ 18}+\frac{ 5}{ 18}\sqrt{ 5}\)\(-\frac{ 11}{ 16}+\frac{ 5}{ 16}\sqrt{ 5}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)
\(1\)\(1\)\(3\)\(0\)\(3\)\(1\)\(1\)\(1\)
\(2\)\(2\)\(4\)\(0\)\(4\)\(2\)\(2\)\(1\)

Note:

Hadamard product $ b \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 $Q(\sqrt{?})$.

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