Summary

You searched for: Spectrum0=1/2,1/2,3/2,3/2

Your search produced 26 matches

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1

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

Degree: 2

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

Maple   LaTex

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: 2.17 |  AESZ: 111  |  Superseeker: 32 1440  |  Hash: d8535e0f3d0bfd4ebcc9c042df43c218  

Degree: 2

\(\theta^4-2^{4} x(2\theta+1)^2(8\theta^2+8\theta+3)+2^{12} x^{2}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 48, 5904, 940800, 169520400, ...
--> OEIS
Normalized instanton numbers (n0=1): 32, -96, 1440, 19704, -14496, ... ; Common denominator:...

Discriminant

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

Local exponents

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

Note:

B-Incarnations:
Fibre product 81111- x 18--21, 4*11-- x 53211,
Double Octics: D.O.8, D.O.36, D.O.73, D.O.249, D.O.258,D.O.265

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3

New Number: 2.1 |  AESZ: 45  |  Superseeker: 12 3204  |  Hash: cdf289f6febf84eb577a238542a57457  

Degree: 2

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

Maple   LaTex

Coefficients of the holomorphic solution: 1, 8, 360, 22400, 1695400, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, 163, 3204, 107582, 4203360, ... ; Common denominator:...

Discriminant

\(-(16z+1)(128z-1)\)

Local exponents

\(-\frac{ 1}{ 16}\)\(0\)\(\frac{ 1}{ 128}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(\frac{ 3}{ 2}\)
\(2\)\(0\)\(2\)\(\frac{ 3}{ 2}\)

Note:

Hadamard product $A \ast a$, where $A$ is (:case 2.1.1)

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4

New Number: 2.20 |  AESZ: 133  |  Superseeker: 12 -3284/3  |  Hash: 4c9628f7dd48f4e9e6ec75303e557389  

Degree: 2

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

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 324, 8400, 44100, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, -42, -3284/3, -20538, -103776, ... ; Common denominator:...

Discriminant

\(1-144z+6912z^2\)

Local exponents

\(0\)\(\frac{ 1}{ 96}-\frac{ 1}{ 288}\sqrt{ 3}I\)\(\frac{ 1}{ 96}+\frac{ 1}{ 288}\sqrt{ 3}I\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(\frac{ 3}{ 2}\)
\(0\)\(2\)\(2\)\(\frac{ 3}{ 2}\)

Note:

Hadamard product A*f
Explicit solution not yet verified

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5

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

Degree: 2

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

Maple   LaTex

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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6

New Number: 2.28 |  AESZ: 49  |  Superseeker: 48 2864  |  Hash: 0a357a8c4fd703ab062148eadcd94daa  

Degree: 2

\(\theta^4-2^{2} 3 x(2\theta+1)^2(18\theta^2+18\theta+7)+2^{4} 3^{6} x^{2}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 84, 17820, 4868400, 1499003100, ...
--> OEIS
Normalized instanton numbers (n0=1): 48, -438, 2864, 77958, -4942032, ... ; Common denominator:...

Discriminant

\((432z-1)^2\)

Local exponents

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

Note:

Hadamard product $A \ast$

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7

New Number: 2.31 |  AESZ: 7**  |  Superseeker: 96 -12064  |  Hash: 2c494088fc85599f73fb344776dbbb28  

Degree: 2

\(\theta^4-2^{4} x(2\theta+1)^2(32\theta^2+32\theta+13)+2^{16} x^{2}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 208, 107280, 70739200, 52362595600, ...
--> OEIS
Normalized instanton numbers (n0=1): 96, -3560, -12064, 1941800, -489007584, ... ; Common denominator:...

Discriminant

\((1024z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 1024}\)\(\infty\)
\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(\frac{ 1}{ 4}\)\(\frac{ 1}{ 2}\)
\(0\)\(\frac{ 3}{ 4}\)\(\frac{ 3}{ 2}\)
\(0\)\(1\)\(\frac{ 3}{ 2}\)

Note:

This operator replaces AESZ 43, to which it is equivalent.

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8

New Number: 2.35 |  AESZ: ~67  |  Superseeker: 480 -16034720  |  Hash: f06ee3928cd6d738db065f3f83d12160  

Degree: 2

\(\theta^4-2^{4} 3 x(2\theta+1)^2(72\theta^2+72\theta+31)+2^{12} 3^{6} x^{2}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 1488, 5351184, 24363091200, 123873273392400, ...
--> OEIS
Normalized instanton numbers (n0=1): 480, -226968, -16034720, 10943202744, -4352645747040, ... ; Common denominator:...

Discriminant

\((6912z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 6912}\)\(\infty\)
\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(\frac{ 1}{ 6}\)\(\frac{ 1}{ 2}\)
\(0\)\(\frac{ 5}{ 6}\)\(\frac{ 3}{ 2}\)
\(0\)\(1\)\(\frac{ 3}{ 2}\)

Note:

This is operator "2.35" from ...

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9

New Number: 2.5 |  AESZ: 25  |  Superseeker: 20 8220  |  Hash: 93279abcbeeade30c29508de7784e582  

Degree: 2

\(\theta^4-2^{2} x(2\theta+1)^2(11\theta^2+11\theta+3)-2^{4} x^{2}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 684, 58800, 6129900, ...
--> OEIS
Normalized instanton numbers (n0=1): 20, 277, 8220, 352994, 18651536, ... ; Common denominator:...

Discriminant

\(1-176z-256z^2\)

Local exponents

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

Note:

Hadamard product $A\ast b$

A-incarnation: X(1,2,2) in G(2,5)

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10

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

Degree: 2

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

Maple   LaTex

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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11

New Number: 4.41 |  AESZ: 220  |  Superseeker: 128 382592  |  Hash: 671a1aa788ead53985e13ad6774d0189  

Degree: 4

\(\theta^4-2^{4} x\left(20\theta^4+56\theta^3+38\theta^2+10\theta+1\right)-2^{10} x^{2}\left(84\theta^4+240\theta^3+261\theta^2+134\theta+25\right)-2^{16} x^{3}(2\theta+1)^2(23\theta^2+55\theta+39)-2^{23} x^{4}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 16, 3600, 851200, 257328400, ...
--> OEIS
Normalized instanton numbers (n0=1): 128, 4084, 382592, 51510860, 8644861312, ... ; Common denominator:...

Discriminant

\(-(512z-1)(64z+1)^3\)

Local exponents

\(-\frac{ 1}{ 64}\)\(0\)\(\frac{ 1}{ 512}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 2}\)\(0\)\(1\)\(\frac{ 1}{ 2}\)
\(\frac{ 3}{ 2}\)\(0\)\(1\)\(\frac{ 3}{ 2}\)
\(2\)\(0\)\(2\)\(\frac{ 3}{ 2}\)

Note:

Sporadic Operator.
Reducible to 3.32, so not a primary operator.
B-Incarnation: 81111- x 82--11

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12

New Number: 5.100 |  AESZ: 347  |  Superseeker: 15 27140/3  |  Hash: f00de20026c099e75b447c475ab287e4  

Degree: 5

\(\theta^4-3 x\left(213\theta^4+186\theta^3+149\theta^2+56\theta+8\right)+2^{3} 3^{3} x^{2}\left(702\theta^4+1078\theta^3+949\theta^2+392\theta+60\right)-2^{6} 3^{3} x^{3}\left(9277\theta^4+18432\theta^3+16008\theta^2+6000\theta+840\right)+2^{13} 3^{4} 5 x^{4}(2\theta+1)^2(51\theta^2+69\theta+32)-2^{14} 3^{6} 5^{2} x^{5}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 24, 1944, 218400, 28488600, ...
--> OEIS
Normalized instanton numbers (n0=1): 15, 1329/4, 27140/3, 220680, 5952570, ... ; Common denominator:...

Discriminant

\(-(192z-1)(1728z^2-207z+1)(-1+120z)^2\)

Local exponents

\(0\)\(\frac{ 23}{ 384}-\frac{ 11}{ 1152}\sqrt{ 33}\)\(\frac{ 1}{ 192}\)\(\frac{ 1}{ 120}\)\(\frac{ 23}{ 384}+\frac{ 11}{ 1152}\sqrt{ 33}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(3\)\(1\)\(\frac{ 3}{ 2}\)
\(0\)\(2\)\(2\)\(4\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.100" from ...

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13

New Number: 5.116 |  AESZ: 414  |  Superseeker: -22432 -425234532128  |  Hash: 973fceefe183415b5d0e15e5a0bd12f5  

Degree: 5

\(\theta^4-2^{4} x\left(8960\theta^4-512\theta^3+736\theta^2+992\theta+183\right)+2^{19} x^{2}\left(14336\theta^4+4352\theta^3+9008\theta^2+2544\theta+261\right)-2^{32} 3^{2} x^{3}\left(4608\theta^4+5632\theta^3+5408\theta^2+2208\theta+351\right)+2^{49} 3^{3} x^{4}(2\theta+1)^2(32\theta^2+48\theta+27)-2^{64} 3^{3} x^{5}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 2928, 21778704, 210543916800, 2314156512099600, ...
--> OEIS
Normalized instanton numbers (n0=1): -22432, -74752296, -425234532128, -3159114140624208, -27288043319514722784, ... ; Common denominator:...

Discriminant

\(-(-1+12288z)(49152z-1)^2(16384z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 49152}\)\(\frac{ 1}{ 16384}\)\(\frac{ 1}{ 12288}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 1}{ 2}\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 3}{ 2}\)
\(0\)\(4\)\(1\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.116" from ...

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14

New Number: 5.128 |  AESZ:  |  Superseeker: -50 -20600  |  Hash: 5a0123bd26e43e2fd9c7e6c3d21a2a33  

Degree: 5

\(\theta^4+2 5 x\left(60\theta^3+45\theta^2+15\theta+2\right)-2^{2} 5^{4} x^{2}\left(8\theta^4+8\theta^3-29\theta^2-20\theta-4\right)-2^{4} 5^{5} x^{3}\left(16\theta^4+216\theta^3+288\theta^2+147\theta+26\right)+2^{6} 5^{7} x^{4}(13\theta^2+37\theta+27)(2\theta+1)^2-2^{8} 5^{9} x^{5}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -20, 900, -38000, 122500, ...
--> OEIS
Normalized instanton numbers (n0=1): -50, -1675/2, -20600, -1433000, -408984396/5, ... ; Common denominator:...

Discriminant

\(-(800000z^3-10000z^2-200z-1)(-1+100z)^2\)

Local exponents

≈\(-0.006091-0.003681I\) ≈\(-0.006091+0.003681I\)\(0\)\(\frac{ 1}{ 100}\) ≈\(0.024681\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(3\)\(1\)\(\frac{ 3}{ 2}\)
\(2\)\(2\)\(0\)\(4\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.128" from ...

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15

New Number: 5.130 |  AESZ:  |  Superseeker: 108 122756  |  Hash: 829aca3d7a00547e299bf794c8643162  

Degree: 5

\(\theta^4-2^{2} 3 x\left(12\theta^4+96\theta^3+71\theta^2+23\theta+3\right)-2^{4} 3^{3} x^{2}\left(160\theta^4+64\theta^3-544\theta^2-340\theta-65\right)+2^{8} 3^{5} x^{3}\left(32\theta^4+576\theta^3+588\theta^2+240\theta+35\right)+2^{12} 3^{7} x^{4}(28\theta^2+52\theta+31)(2\theta+1)^2+2^{16} 3^{9} x^{5}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 36, 3780, 558000, 98828100, ...
--> OEIS
Normalized instanton numbers (n0=1): 108, -1782, 122756, -5930658, 607239072, ... ; Common denominator:...

Discriminant

\((144z-1)(6912z^2+288z-1)(1+144z)^2\)

Local exponents

\(-\frac{ 1}{ 48}-\frac{ 1}{ 72}\sqrt{ 3}\)\(-\frac{ 1}{ 144}\)\(0\)\(-\frac{ 1}{ 48}+\frac{ 1}{ 72}\sqrt{ 3}\)\(\frac{ 1}{ 144}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(3\)\(0\)\(1\)\(1\)\(\frac{ 3}{ 2}\)
\(2\)\(4\)\(0\)\(2\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.130" from ...

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16

New Number: 5.29 |  AESZ: 208  |  Superseeker: 274/7 281388/7  |  Hash: f1d6dfa8a5cdcc2513dfca4243565b2f  

Degree: 5

\(7^{2} \theta^4-2 7 x\left(1056\theta^4+1884\theta^3+1397\theta^2+455\theta+56\right)+2^{2} 3 x^{2}\left(22760\theta^4+13672\theta^3-22537\theta^2-18116\theta-3584\right)-2^{4} x^{3}\left(53312\theta^4-162120\theta^3-195172\theta^2-78561\theta-11130\right)-2^{6} 19 x^{4}(1189\theta^2+2533\theta+1646)(2\theta+1)^2+2^{11} 19^{2} x^{5}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 16, 1440, 196000, 32418400, ...
--> OEIS
Normalized instanton numbers (n0=1): 274/7, 6115/7, 281388/7, 2815228, 1699166270/7, ... ; Common denominator:...

Discriminant

\((4z+1)(512z^2-284z+1)(-7+76z)^2\)

Local exponents

\(-\frac{ 1}{ 4}\)\(0\)\(\frac{ 71}{ 256}-\frac{ 17}{ 256}\sqrt{ 17}\)\(\frac{ 7}{ 76}\)\(\frac{ 71}{ 256}+\frac{ 17}{ 256}\sqrt{ 17}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(3\)\(1\)\(\frac{ 3}{ 2}\)
\(2\)\(0\)\(2\)\(4\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.29" from ...

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17

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

Degree: 5

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

Maple   LaTex

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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18

New Number: 5.36 |  AESZ: 219  |  Superseeker: 166/5 360988/15  |  Hash: b7068bb339f61ebd7c591b7be3fe5893  

Degree: 5

\(5^{2} \theta^4-2 5 x\left(464\theta^4+1036\theta^3+763\theta^2+245\theta+30\right)-2^{2} 3^{2} x^{2}\left(7064\theta^4+22472\theta^3+26699\theta^2+13200\theta+2340\right)-2^{4} 3^{4} x^{3}\left(3440\theta^4+13320\theta^3+18784\theta^2+10665\theta+2070\right)-2^{6} 3^{8} x^{4}(19\theta^2+59\theta+45)(2\theta+1)^2-2^{8} 3^{9} x^{5}(2\theta+1)^2(2\theta+3)^2\)

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Coefficients of the holomorphic solution: 1, 12, 972, 109200, 14949900, ...
--> OEIS
Normalized instanton numbers (n0=1): 166/5, 638, 360988/15, 7222128/5, 524377242/5, ... ; Common denominator:...

Discriminant

\(-(16z+1)(3888z^2+216z-1)(5+36z)^2\)

Local exponents

\(-\frac{ 5}{ 36}\)\(-\frac{ 1}{ 16}\)\(-\frac{ 1}{ 36}-\frac{ 1}{ 54}\sqrt{ 3}\)\(0\)\(-\frac{ 1}{ 36}+\frac{ 1}{ 54}\sqrt{ 3}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(\frac{ 1}{ 2}\)
\(3\)\(1\)\(1\)\(0\)\(1\)\(\frac{ 3}{ 2}\)
\(4\)\(2\)\(2\)\(0\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.36" from ...

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19

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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20

New Number: 5.41 |  AESZ: 230  |  Superseeker: 291 7935104  |  Hash: ac762b013587176079179af09a110ab6  

Degree: 5

\(\theta^4+3 x\left(945\theta^4-162\theta^3-49\theta^2+32\theta+8\right)+2 3^{4} x^{2}\left(17928\theta^4+2970\theta^3+10187\theta^2+3376\theta+408\right)+2^{2} 3^{7} x^{3}\left(156285\theta^4+200016\theta^3+193630\theta^2+84378\theta+13964\right)+2^{4} 3^{10} 19 x^{4}(4743\theta^2+8199\theta+4922)(2\theta+1)^2+2^{9} 3^{15} 19^{2} x^{5}(2\theta+1)^2(2\theta+3)^2\)

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Coefficients of the holomorphic solution: 1, -24, -648, 494400, -82643400, ...
--> OEIS
Normalized instanton numbers (n0=1): 291, -38832, 7935104, -2098976940, 651305429796, ... ; Common denominator:...

Discriminant

\((432z+1)(93312z^2+351z+1)(1+1026z)^2\)

Local exponents

\(-\frac{ 1}{ 432}\)\(-\frac{ 13}{ 6912}-\frac{ 7}{ 6912}\sqrt{ 7}I\)\(-\frac{ 13}{ 6912}+\frac{ 7}{ 6912}\sqrt{ 7}I\)\(-\frac{ 1}{ 1026}\)\(0\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(1\)\(1\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(1\)\(3\)\(0\)\(\frac{ 3}{ 2}\)
\(2\)\(2\)\(2\)\(4\)\(0\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.41" from ...

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21

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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22

New Number: 5.53 |  AESZ: 259  |  Superseeker: 82450 22323908689400  |  Hash: 8b20756bb52131d41c44fd699c9e3a24  

Degree: 5

\(\theta^4+2 5 x\left(40000\theta^4-17500\theta^3-8125\theta^2+625\theta+238\right)+2^{2} 5^{6} x^{2}\left(835000\theta^4-365000\theta^3+371125\theta^2+58500\theta+2116\right)+2^{4} 5^{11} x^{3}\left(3130000\theta^4+1815000\theta^3+1662000\theta^2+625875\theta+96914\right)+2^{6} 5^{19} 13 x^{4}(625\theta^2+745\theta+351)(2\theta+1)^2+2^{8} 5^{25} 13^{2} x^{5}(2\theta+1)^2(2\theta+3)^2\)

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Coefficients of the holomorphic solution: 1, -2380, 14400900, -112575082000, 993749164922500, ...
--> OEIS
Normalized instanton numbers (n0=1): 82450, -976323150, 22323908689400, -680892969306394000, 24398212781075814030620, ... ; Common denominator:...

Discriminant

\((1+50000z)(12500z+1)^2(162500z+1)^2\)

Local exponents

\(-\frac{ 1}{ 12500}\)\(-\frac{ 1}{ 50000}\)\(-\frac{ 1}{ 162500}\)\(0\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 2}\)\(1\)\(1\)\(0\)\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 2}\)\(1\)\(3\)\(0\)\(\frac{ 3}{ 2}\)
\(1\)\(2\)\(4\)\(0\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.53" from ...

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23

New Number: 5.54 |  AESZ: 260  |  Superseeker: -188/5 -450516/5  |  Hash: 03ff8e2e94b897c3891e6981e7fb4ec9  

Degree: 5

\(5^{2} \theta^4+2^{2} 5 x\left(596\theta^4+544\theta^3+397\theta^2+125\theta+15\right)+2^{4} 3 x^{2}\left(30048\theta^4+14784\theta^3-13312\theta^2-10940\theta-2115\right)+2^{8} 3^{3} x^{3}\left(6368\theta^4-6720\theta^3-9052\theta^2-4080\theta-655\right)-2^{12} 3^{6} x^{4}(2\theta+1)^2(76\theta^2+196\theta+139)-2^{16} 3^{9} x^{5}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -12, 1260, -188400, 34353900, ...
--> OEIS
Normalized instanton numbers (n0=1): -188/5, 7693/5, -450516/5, 37785946/5, -790482672, ... ; Common denominator:...

Discriminant

\(-(16z+1)(6912z^2-288z-1)(5+432z)^2\)

Local exponents

\(-\frac{ 1}{ 16}\)\(-\frac{ 5}{ 432}\)\(\frac{ 1}{ 48}-\frac{ 1}{ 72}\sqrt{ 3}\)\(0\)\(\frac{ 1}{ 48}+\frac{ 1}{ 72}\sqrt{ 3}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(3\)\(1\)\(0\)\(1\)\(\frac{ 3}{ 2}\)
\(2\)\(4\)\(2\)\(0\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.54" from ...

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24

New Number: 5.74 |  AESZ: 297  |  Superseeker: 26/7 55644/7  |  Hash: cd0b6008fa6b70d89e004100b5698063  

Degree: 5

\(7^{2} \theta^4-2 7 x\theta(520\theta^3+68\theta^2+41\theta+7)-2^{2} 3 x^{2}\left(9480\theta^4+153912\theta^3+212893\theta^2+108080\theta+18816\right)+2^{4} 3^{3} 7 x^{3}\left(13424\theta^4+48792\theta^3+45656\theta^2+17979\theta+2606\right)-2^{6} 3^{7} x^{4}(2\theta+1)^2(2257\theta^2+3601\theta+1942)+2^{11} 3^{11} x^{5}(2\theta+1)^2(2\theta+3)^2\)

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Coefficients of the holomorphic solution: 1, 0, 288, 7200, 1058400, ...
--> OEIS
Normalized instanton numbers (n0=1): 26/7, 2594/7, 55644/7, 2996576/7, 135364470/7, ... ; Common denominator:...

Discriminant

\((128z-1)(432z^2-72z-1)(-7+324z)^2\)

Local exponents

\(\frac{ 1}{ 12}-\frac{ 1}{ 18}\sqrt{ 3}\)\(0\)\(\frac{ 1}{ 128}\)\(\frac{ 7}{ 324}\)\(\frac{ 1}{ 12}+\frac{ 1}{ 18}\sqrt{ 3}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(3\)\(1\)\(\frac{ 3}{ 2}\)
\(2\)\(0\)\(2\)\(4\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.74" from ...

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25

New Number: 5.83 |  AESZ: 316  |  Superseeker: 852/11 1678156/11  |  Hash: b8201d587a016cc013e2477aadb5c1ff  

Degree: 5

\(11^{2} \theta^4-2^{2} 3 11 x\left(364\theta^4+824\theta^3+599\theta^2+187\theta+22\right)-2^{5} x^{2}\left(62164\theta^4+84496\theta^3+12499\theta^2-6402\theta-1584\right)-2^{4} 3 x^{3}\left(484016\theta^4+474144\theta^3+366952\theta^2+161832\theta+27027\right)-2^{11} 3^{2} x^{4}(964\theta^2+1360\theta+669)(2\theta+1)^2-2^{16} 3^{4} x^{5}(2\theta+1)^2(2\theta+3)^2\)

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Coefficients of the holomorphic solution: 1, 24, 3240, 675600, 171901800, ...
--> OEIS
Normalized instanton numbers (n0=1): 852/11, 21572/11, 1678156/11, 15912512, 22956446184/11, ... ; Common denominator:...

Discriminant

\(-(2304z^3+1664z^2+432z-1)(11+192z)^2\)

Local exponents

≈\(-0.362258-0.240689I\) ≈\(-0.362258+0.240689I\)\(-\frac{ 11}{ 192}\)\(0\) ≈\(0.002294\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(3\)\(0\)\(1\)\(\frac{ 3}{ 2}\)
\(2\)\(2\)\(4\)\(0\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.83" from ...

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26

New Number: 8.83 |  AESZ:  |  Superseeker: 208 642704  |  Hash: 7314ca8e48f991223dc4e1c8b4893b95  

Degree: 8

\(\theta^4-2^{4} x\left(116\theta^4+160\theta^3+119\theta^2+39\theta+5\right)+2^{9} x^{2}\left(2096\theta^4+5600\theta^3+5694\theta^2+2366\theta+355\right)-2^{15} x^{3}\left(4232\theta^4+22416\theta^3+28566\theta^2+11646\theta+1745\right)-2^{21} x^{4}\left(20616\theta^4+8496\theta^3-69074\theta^2-48074\theta-9335\right)+2^{27} x^{5}\left(49408\theta^4+114208\theta^3-29684\theta^2-42372\theta-9585\right)+2^{34} x^{6}\left(46496\theta^4-21984\theta^3-28956\theta^2-5580\theta+375\right)-2^{41} 5 x^{7}(2\theta+1)^2(344\theta^2+416\theta+163)-2^{48} 5^{2} x^{8}(2\theta+1)^2(2\theta+3)^2\)

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Coefficients of the holomorphic solution: 1, 80, 23760, 9900800, 4805155600, ...
--> OEIS
Normalized instanton numbers (n0=1): 208, 3154, 642704, -4424361, 3864242160, ... ; Common denominator:...

Discriminant

\(-(16384z^2-768z+1)(4096z^2+704z-1)(128z+1)^2(320z-1)^2\)

Local exponents

\(-\frac{ 11}{ 128}-\frac{ 5}{ 128}\sqrt{ 5}\)\(-\frac{ 1}{ 128}\)\(0\)\(\frac{ 3}{ 128}-\frac{ 1}{ 64}\sqrt{ 2}\)\(-\frac{ 11}{ 128}+\frac{ 5}{ 128}\sqrt{ 5}\)\(\frac{ 1}{ 320}\)\(\frac{ 3}{ 128}+\frac{ 1}{ 64}\sqrt{ 2}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(3\)\(0\)\(1\)\(1\)\(3\)\(1\)\(\frac{ 3}{ 2}\)
\(2\)\(4\)\(0\)\(2\)\(2\)\(4\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "8.83" from ...

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