Summary

You searched for: h3=16

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1

New Number: 2.15 |  AESZ: 38  |  Superseeker: 48 73328  |  Hash: 9ce26bb7405c3b98d8aeae5b1102c611  

Degree: 2

\(\theta^4-2^{4} x(4\theta+1)(4\theta+3)(3\theta^2+3\theta+1)+2^{9} x^{2}(4\theta+1)(4\theta+3)(4\theta+5)(4\theta+7)\)

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Coefficients of the holomorphic solution: 1, 48, 8400, 2069760, 609008400, ...
--> OEIS
Normalized instanton numbers (n0=1): 48, 998, 73328, 7388135, 857248528, ... ; Common denominator:...

Discriminant

\((512z-1)(256z-1)\)

Local exponents

\(0\)\(\frac{ 1}{ 512}\)\(\frac{ 1}{ 256}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 4}\)
\(0\)\(1\)\(1\)\(\frac{ 3}{ 4}\)
\(0\)\(1\)\(1\)\(\frac{ 5}{ 4}\)
\(0\)\(2\)\(2\)\(\frac{ 7}{ 4}\)

Note:

Hadamard product $C\ast d$

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2

New Number: 4.45 |  AESZ: 233  |  Superseeker: 80 104976  |  Hash: 03f67459f6d678669f766c99281b1e79  

Degree: 4

\(\theta^4-2^{4} x\left(83\theta^4+94\theta^3+71\theta^2+24\theta+3\right)+2^{11} 3 x^{2}\left(101\theta^4+191\theta^3+174\theta^2+71\theta+10\right)-2^{16} 3^{2} x^{3}\left(203\theta^4+432\theta^3+333\theta^2+102\theta+11\right)+2^{23} 3^{3} x^{4}(3\theta+1)(2\theta+1)^2(3\theta+2)\)

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Coefficients of the holomorphic solution: 1, 48, 9360, 2553600, 813027600, ...
--> OEIS
Normalized instanton numbers (n0=1): 80, 2794, 104976, 5367454, 508265072, ... ; Common denominator:...

Discriminant

\((512z-1)(432z-1)(-1+192z)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 512}\)\(\frac{ 1}{ 432}\)\(\frac{ 1}{ 192}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 3}\)
\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(3\)\(\frac{ 1}{ 2}\)
\(0\)\(2\)\(2\)\(4\)\(\frac{ 2}{ 3}\)

Note:

Sporadic Operator.

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3

New Number: 5.101 |  AESZ: 348  |  Superseeker: -52 -44772  |  Hash: 8759f016475d17d0fc88f4b98a374d3f  

Degree: 5

\(\theta^4+2^{2} x\left(70\theta^4+194\theta^3+145\theta^2+48\theta+6\right)-2^{4} 3 x^{2}\left(141\theta^4-858\theta^3-2111\theta^2-1192\theta-206\right)-2^{8} 3^{2} x^{3}\left(18\theta^4-324\theta^3-2364\theta^2-1953\theta-403\right)-2^{10} 3^{4} x^{4}(3\theta+1)(3\theta+2)(42\theta^2+258\theta+223)+2^{14} 3^{6} x^{5}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)\)

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Coefficients of the holomorphic solution: 1, -24, 2160, -309120, 54608400, ...
--> OEIS
Normalized instanton numbers (n0=1): -52, 461/2, -44772, 3546761/2, -178670332, ... ; Common denominator:...

Discriminant

\((746496z^3+17280z^2+352z+1)(-1+36z)^2\)

Local exponents

≈\(-0.009925-0.017537I\) ≈\(-0.009925+0.017537I\) ≈\(-0.003299\)\(0\)\(\frac{ 1}{ 36}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 3}\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(\frac{ 2}{ 3}\)
\(1\)\(1\)\(1\)\(0\)\(3\)\(\frac{ 4}{ 3}\)
\(2\)\(2\)\(2\)\(0\)\(4\)\(\frac{ 5}{ 3}\)

Note:

This is operator "5.101" from ...

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4

New Number: 5.14 |  AESZ: 116  |  Superseeker: 64 23360  |  Hash: 0b366ad8c78b6697205c5a7fff270f5b  

Degree: 5

\(\theta^4-2^{5} x\left(10\theta^4+26\theta^3+20\theta^2+7\theta+1\right)+2^{8} x^{2}\left(52\theta^4+472\theta^3+832\theta^2+492\theta+103\right)+2^{16} x^{3}\left(14\theta^4+12\theta^3-96\theta^2-105\theta-29\right)-2^{18} x^{4}(2\theta+1)(56\theta^3+468\theta^2+646\theta+249)-2^{24} x^{5}(2\theta+1)(4\theta+3)(4\theta+5)(2\theta+3)\)

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Coefficients of the holomorphic solution: 1, 32, 2448, 273920, 38525200, ...
--> OEIS
Normalized instanton numbers (n0=1): 64, 12, 23360, 654490, 53956288, ... ; Common denominator:...

Discriminant

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

Local exponents

\(-\frac{ 1}{ 32}\)\(0\)\(\frac{ 1}{ 256}\)\(\frac{ 1}{ 64}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(\frac{ 1}{ 2}\)\(\frac{ 3}{ 4}\)
\(3\)\(0\)\(1\)\(\frac{ 1}{ 2}\)\(\frac{ 5}{ 4}\)
\(4\)\(0\)\(2\)\(1\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.14" from ...

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5

New Number: 5.73 |  AESZ: 293  |  Superseeker: 20 13188  |  Hash: f19eeaee48396d15d7cf7be47d7d48a7  

Degree: 5

\(\theta^4-2^{2} x\left(54\theta^4+66\theta^3+49\theta^2+16\theta+2\right)+2^{4} x^{2}\left(417\theta^4-306\theta^3-1219\theta^2-776\theta-154\right)+2^{8} x^{3}\left(166\theta^4+1920\theta^3+1589\theta^2+432\theta+23\right)-2^{12} 7 x^{4}(2\theta+1)(38\theta^3+45\theta^2+12\theta-2)-2^{14} 7^{2} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)\)

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Coefficients of the holomorphic solution: 1, 8, 528, 45440, 4763920, ...
--> OEIS
Normalized instanton numbers (n0=1): 20, 867/2, 13188, 609734, 35512476, ... ; Common denominator:...

Discriminant

\(-(16z+1)(256z^2+176z-1)(-1+28z)^2\)

Local exponents

\(-\frac{ 11}{ 32}-\frac{ 5}{ 32}\sqrt{ 5}\)\(-\frac{ 1}{ 16}\)\(0\)\(-\frac{ 11}{ 32}+\frac{ 5}{ 32}\sqrt{ 5}\)\(\frac{ 1}{ 28}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)
\(1\)\(1\)\(0\)\(1\)\(3\)\(1\)
\(2\)\(2\)\(0\)\(2\)\(4\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.73" from ...

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6

New Number: 5.89 |  AESZ: 329  |  Superseeker: 48 25200  |  Hash: 8c526b3b825d5ad6a3d0fb83ee4e6059  

Degree: 5

\(\theta^4-2^{4} x\left(8\theta^4+34\theta^3+25\theta^2+8\theta+1\right)-2^{8} x^{2}\left(87\theta^4+150\theta^3+32\theta^2-2\theta-1\right)-2^{12} x^{3}\left(202\theta^4+240\theta^3+211\theta^2+102\theta+19\right)-2^{16} 3 x^{4}(2\theta+1)(22\theta^3+45\theta^2+38\theta+12)-2^{20} 3^{2} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 16, 1200, 136960, 19010320, ...
--> OEIS
Normalized instanton numbers (n0=1): 48, 270, 25200, 968066, 80892688, ... ; Common denominator:...

Discriminant

\(-(16384z^3+3072z^2+224z-1)(1+48z)^2\)

Local exponents

≈\(-0.095858-0.072741I\) ≈\(-0.095858+0.072741I\)\(-\frac{ 1}{ 48}\)\(0\) ≈\(0.004215\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)
\(1\)\(1\)\(3\)\(0\)\(1\)\(1\)
\(2\)\(2\)\(4\)\(0\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.89" from ...

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7

New Number: 5.90 |  AESZ: 330  |  Superseeker: 352 3284448  |  Hash: ba5b66d5fe92237e6416a117563571e9  

Degree: 5

\(\theta^4+2^{4} x\left(112\theta^4-64\theta^3-32\theta^2+1\right)+2^{14} x^{2}\left(56\theta^4-64\theta^3+3\theta^2-10\theta-4\right)+2^{20} x^{3}\left(32\theta^4-384\theta^3-436\theta^2-264\theta-55\right)-2^{29} 3 x^{4}(2\theta+1)(10\theta+7)(2\theta^2+4\theta+3)-2^{38} 3^{2} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)\)

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Coefficients of the holomorphic solution: 1, -16, 4368, -344320, 107445520, ...
--> OEIS
Normalized instanton numbers (n0=1): 352, -23368, 3284448, -578330224, 120252731680, ... ; Common denominator:...

Discriminant

\(-(-1+256z)(256z+1)^2(768z+1)^2\)

Local exponents

\(-\frac{ 1}{ 256}\)\(-\frac{ 1}{ 768}\)\(0\)\(\frac{ 1}{ 256}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 2}\)\(1\)\(0\)\(1\)\(1\)
\(\frac{ 1}{ 2}\)\(3\)\(0\)\(1\)\(1\)
\(1\)\(4\)\(0\)\(2\)\(\frac{ 3}{ 2}\)

Note:

B-Incarnation as double octic D.O.20

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8

New Number: 5.91 |  AESZ: 331  |  Superseeker: 112 186800  |  Hash: a30093d8c1ab2f66122cef8935b79efb  

Degree: 5

\(\theta^4+2^{4} x\left(18\theta^4-48\theta^3-33\theta^2-9\theta-1\right)-2^{9} x^{2}\left(86\theta^4+512\theta^3+125\theta^2+45\theta+10\right)-2^{14} x^{3}\left(1138\theta^4+2040\theta^3+1883\theta^2+879\theta+157\right)-2^{19} 7 x^{4}(2\theta+1)(186\theta^3+375\theta^2+317\theta+100)-2^{27} 7^{2} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 16, 1488, 183040, 27611920, ...
--> OEIS
Normalized instanton numbers (n0=1): 112, -2242, 186800, -11675813, 1250599376, ... ; Common denominator:...

Discriminant

\(-(32z+1)(256z-1)(64z+1)(1+224z)^2\)

Local exponents

\(-\frac{ 1}{ 32}\)\(-\frac{ 1}{ 64}\)\(-\frac{ 1}{ 224}\)\(0\)\(\frac{ 1}{ 256}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)
\(1\)\(1\)\(3\)\(0\)\(1\)\(1\)
\(2\)\(2\)\(4\)\(0\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.91" from ...

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9

New Number: 1.3 |  AESZ: 3  |  Superseeker: 32 26016  |  Hash: e7a9c334fb603aceccc0517dab63e7d4  

Degree: 1

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

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Coefficients of the holomorphic solution: 1, 16, 1296, 160000, 24010000, ...
--> OEIS
Normalized instanton numbers (n0=1): 32, 608, 26016, 1606496, 122373984, ... ; Common denominator:...

Discriminant

\(\)

No data for singularities

Note:

A-incarnation: X(2,2,2,2) in P^7.

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