Summary

You searched for: c2h=48

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1

New Number: 4.33 |  AESZ: 55  |  Superseeker: 76/3 144196/3  |  Hash: 7e88cd5b7dc1c51022b66ac6f009218f  

Degree: 4

\(3^{2} \theta^4-2^{2} 3 x\left(208\theta^4+224\theta^3+163\theta^2+51\theta+6\right)+2^{9} x^{2}\left(32\theta^4-928\theta^3-1606\theta^2-837\theta-141\right)+2^{16} x^{3}\left(144\theta^4+576\theta^3+467\theta^2+144\theta+15\right)-2^{24} x^{4}\left((2\theta+1)^4\right)\)

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Coefficients of the holomorphic solution: 1, 8, 936, 108800, 16748200, ...
--> OEIS
Normalized instanton numbers (n0=1): 76/3, 3476/3, 144196/3, 3563196, 309069600, ... ; Common denominator:...

Discriminant

\(-(64z+1)(256z-1)(-3+128z)^2\)

Local exponents

\(-\frac{ 1}{ 64}\)\(0\)\(\frac{ 1}{ 256}\)\(\frac{ 3}{ 128}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(3\)\(\frac{ 1}{ 2}\)
\(2\)\(0\)\(2\)\(4\)\(\frac{ 1}{ 2}\)

Note:

Sporadic operator. There is a second MUM-point
hiding at infinity, corresponding to Operator 4.56

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2

New Number: 5.32 |  AESZ: 215  |  Superseeker: 220/3 89212  |  Hash: ced61f5675491a3c4446c0e55e7bc36b  

Degree: 5

\(3^{2} \theta^4-2^{2} 3 x\left(268\theta^4+632\theta^3+463\theta^2+147\theta+18\right)-2^{7} x^{2}\left(448\theta^4-1616\theta^3-4280\theta^2-2418\theta-441\right)+2^{12} x^{3}\left(416\theta^4+2016\theta^3+756\theta^2-288\theta-135\right)+2^{19} x^{4}(8\theta^2-28\theta-33)(2\theta+1)^2-2^{24} x^{5}(2\theta+1)^2(2\theta+3)^2\)

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Coefficients of the holomorphic solution: 1, 24, 2664, 470400, 102047400, ...
--> OEIS
Normalized instanton numbers (n0=1): 220/3, 3538/3, 89212, 7484350, 2459418080/3, ... ; Common denominator:...

Discriminant

\(-(16z-1)(4096z^2-384z+1)(3+64z)^2\)

Local exponents

\(-\frac{ 3}{ 64}\)\(0\)\(\frac{ 3}{ 64}-\frac{ 1}{ 32}\sqrt{ 2}\)\(\frac{ 1}{ 16}\)\(\frac{ 3}{ 64}+\frac{ 1}{ 32}\sqrt{ 2}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(3\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 3}{ 2}\)
\(4\)\(0\)\(2\)\(2\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.32" from ...

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3

New Number: 5.33 |  AESZ: 216  |  Superseeker: 9 14201/3  |  Hash: af7027bf24acce4fd0ed5b09e575e2a5  

Degree: 5

\(\theta^4-3 x\theta(2+11\theta+18\theta^2+27\theta^3)-2 3^{3} x^{2}\left(72\theta^4+414\theta^3+603\theta^2+330\theta+64\right)+2^{2} 3^{5} x^{3}\left(93\theta^4-720\theta^2-708\theta-184\right)+2^{3} 3^{7} x^{4}(2\theta+1)(54\theta^3+405\theta^2+544\theta+200)-2^{4} 3^{10} x^{5}(2\theta+1)(3\theta+2)(3\theta+4)(2\theta+3)\)

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Coefficients of the holomorphic solution: 1, 0, 216, 7200, 567000, ...
--> OEIS
Normalized instanton numbers (n0=1): 9, 225, 14201/3, 154800, 6298596, ... ; Common denominator:...

Discriminant

\(-(27z+1)(108z-1)(36z+1)(-1+18z)^2\)

Local exponents

\(-\frac{ 1}{ 27}\)\(-\frac{ 1}{ 36}\)\(0\)\(\frac{ 1}{ 108}\)\(\frac{ 1}{ 18}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(\frac{ 2}{ 3}\)
\(1\)\(1\)\(0\)\(1\)\(3\)\(\frac{ 4}{ 3}\)
\(2\)\(2\)\(0\)\(2\)\(4\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.33" from ...

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4

New Number: 5.42 |  AESZ: 231  |  Superseeker: 460/3 894404/3  |  Hash: 6f793238336123adfdcd7ee17d64e5ec  

Degree: 5

\(3^{2} \theta^4-2^{2} 3 x\left(28\theta^4+1016\theta^3+739\theta^2+231\theta+30\right)-2^{9} x^{2}\left(1168\theta^4-968\theta^3-9518\theta^2-5325\theta-1005\right)+2^{16} x^{3}\left(988\theta^4+8208\theta^3-743\theta^2-4230\theta-1245\right)+2^{24} 5 x^{4}(2\theta+1)^2(9\theta^2-279\theta-277)-2^{33} 5^{2} x^{5}(2\theta+1)^2(2\theta+3)^2\)

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Coefficients of the holomorphic solution: 1, 40, 3240, 313600, 39327400, ...
--> OEIS
Normalized instanton numbers (n0=1): 460/3, -16828/3, 894404/3, -42271624/3, 2076730720/3, ... ; Common denominator:...

Discriminant

\(-(256z-1)(32768z^2-208z+1)(3+640z)^2\)

Local exponents

\(-\frac{ 3}{ 640}\)\(0\)\(\frac{ 13}{ 4096}-\frac{ 7}{ 4096}\sqrt{ 7}I\)\(\frac{ 13}{ 4096}+\frac{ 7}{ 4096}\sqrt{ 7}I\)\(\frac{ 1}{ 256}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(3\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 3}{ 2}\)
\(4\)\(0\)\(2\)\(2\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.42" from ...

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5

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

Degree: 8

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

Maple   LaTex

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 $Q(\sqrt{?})$.

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6

New Number: 1.11 |  AESZ: 11  |  Superseeker: 324 10792428  |  Hash: 8ac8b98b80383c9f0ea125ccd6e6a55d  

Degree: 1

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

Maple   LaTex

Coefficients of the holomorphic solution: 1, 72, 37800, 31046400, 31216185000, ...
--> OEIS
Normalized instanton numbers (n0=1): 324, 37260, 10792428, 4580482284, 2405245303584, ... ; Common denominator:...

Discriminant

\(1-1728z\)

Local exponents

\(0\)\(\frac{ 1}{ 1728}\)\(\infty\)
\(0\)\(0\)\(\frac{ 1}{ 4}\)
\(0\)\(1\)\(\frac{ 1}{ 3}\)
\(0\)\(1\)\(\frac{ 2}{ 3}\)
\(0\)\(2\)\(\frac{ 3}{ 4}\)

Note:

A-incarnation: X(4,6) in P^5(1,1,1,2,2,3)

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