Summary

You searched for: h3=48

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1

New Number: 2.24 |  AESZ: 137  |  Superseeker: 20 1684/3  |  Hash: 198d6c822d6c46225ac2553d60df6539  

Degree: 2

\(\theta^4-2^{2} x(2\theta+1)^2(17\theta^2+17\theta+6)+2^{7} 3^{2} x^{2}(2\theta+1)^2(2\theta+3)^2\)

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Coefficients of the holomorphic solution: 1, 24, 1512, 124800, 11730600, ...
--> OEIS
Normalized instanton numbers (n0=1): 20, 2, 1684/3, 7602, 173472, ... ; Common denominator:...

Discriminant

\((144z-1)(128z-1)\)

Local exponents

\(0\)\(\frac{ 1}{ 144}\)\(\frac{ 1}{ 128}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(\frac{ 3}{ 2}\)
\(0\)\(2\)\(2\)\(\frac{ 3}{ 2}\)

Note:

Hadamard product $A \ast g$.

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2

New Number: 2.52 |  AESZ: 16  |  Superseeker: 4 644/3  |  Hash: 05af0662662bfbec63e3186c4f363313  

Degree: 2

\(\theta^4-2^{2} x(2\theta+1)^2(5\theta^2+5\theta+2)+2^{8} x^{2}(2\theta+1)(\theta+1)^2(2\theta+3)\)

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Coefficients of the holomorphic solution: 1, 8, 168, 5120, 190120, ...
--> OEIS
Normalized instanton numbers (n0=1): 4, 20, 644/3, 3072, 52512, ... ; Common denominator:...

Discriminant

\((64z-1)(16z-1)\)

Local exponents

\(0\)\(\frac{ 1}{ 64}\)\(\frac{ 1}{ 16}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(1\)
\(0\)\(1\)\(1\)\(1\)
\(0\)\(2\)\(2\)\(\frac{ 3}{ 2}\)

Note:

Hadamard product $I \ast \alpha$
A-Incarnation: diagonal subfamily of 1,1,1,1-intersection in $P^1 \times P^1 \times P^1 \times \P^1$
B-Incarnations:
Fibre products: 62211- x 632--1, S62211

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3

New Number: 5.49 |  AESZ: 248  |  Superseeker: 7/3 148  |  Hash: 0c9ccff1cb4f5096e455a9026799ed5a  

Degree: 5

\(3^{2} \theta^4-3 x\left(106\theta^4+146\theta^3+115\theta^2+42\theta+6\right)-x^{2}\left(4511\theta^4+24314\theta^3+37829\theta^2+23598\theta+5286\right)+2^{2} x^{3}\left(10457\theta^4+32184\theta^3+24449\theta^2+3627\theta-1317\right)-2^{2} 11 x^{4}\left(1596\theta^4+2040\theta^3-101\theta^2-1085\theta-386\right)-2^{4} 11^{2} x^{5}(4\theta+3)(\theta+1)^2(4\theta+5)\)

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Coefficients of the holomorphic solution: 1, 2, 54, 1028, 29110, ...
--> OEIS
Normalized instanton numbers (n0=1): 7/3, 551/24, 148, 8241/4, 86854/3, ... ; Common denominator:...

Discriminant

\(-(16z+1)(16z^2+44z-1)(-3+11z)^2\)

Local exponents

\(-\frac{ 11}{ 8}-\frac{ 5}{ 8}\sqrt{ 5}\)\(-\frac{ 1}{ 16}\)\(0\)\(-\frac{ 11}{ 8}+\frac{ 5}{ 8}\sqrt{ 5}\)\(\frac{ 3}{ 11}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 3}{ 4}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)
\(1\)\(1\)\(0\)\(1\)\(3\)\(1\)
\(2\)\(2\)\(0\)\(2\)\(4\)\(\frac{ 5}{ 4}\)

Note:

This is operator "5.49" from ...

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4

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

Degree: 8

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

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

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5

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

Degree: 8

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

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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 ...

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6

New Number: 8.3 |  AESZ: 105  |  Superseeker: 8 -104  |  Hash: 7b27135451cf2016217211c633b7ab83  

Degree: 8

\(\theta^4-2^{2} x(3\theta^2+3\theta+1)(7\theta^2+7\theta+2)+2^{5} 3 x^{2}\left(15\theta^4+28\theta^3+39\theta^2+22\theta+4\right)-2^{10} x^{3}\left(21\theta^4-126\theta^3-386\theta^2-291\theta-76\right)+2^{14} x^{4}\left(37\theta^4+74\theta^3+50\theta^2+13\theta+6\right)+2^{18} x^{5}\left(21\theta^4+210\theta^3+118\theta^2-19\theta-24\right)+2^{21} 3 x^{6}\left(15\theta^4+32\theta^3+45\theta^2+32\theta+8\right)+2^{26} x^{7}(3\theta^2+3\theta+1)(7\theta^2+7\theta+2)+2^{32} x^{8}\left((\theta+1)^4\right)\)

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Coefficients of the holomorphic solution: 1, 8, 200, 6272, 233896, ...
--> OEIS
Normalized instanton numbers (n0=1): 8, 71/2, -104, 4202, 50112, ... ; Common denominator:...

Discriminant

\((8z+1)(64z-1)(4z+1)(32z-1)(1+256z^2)^2\)

Local exponents

\(-\frac{ 1}{ 4}\)\(-\frac{ 1}{ 8}\)\(0-\frac{ 1}{ 16}I\)\(0\)\(0+\frac{ 1}{ 16}I\)\(\frac{ 1}{ 64}\)\(\frac{ 1}{ 32}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)
\(1\)\(1\)\(3\)\(0\)\(3\)\(1\)\(1\)\(1\)
\(2\)\(2\)\(4\)\(0\)\(4\)\(2\)\(2\)\(1\)

Note:

Hadamard product $a \ast d$. 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
$Q(\sqrt{-1})$.

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