Summary

You searched for: Spectrum0=1,1,1,1

Your search produced 113 matches
 1-30  31-60  61-90  91-113 

You can download all data as plain text or as JSON

61

New Number: 8.18 |  AESZ: 197  |  Superseeker: 3 1621/13  |  Hash: 4cc8bdba73e5fa6cb4089fa5296429de  

Degree: 8

\(13^{2} \theta^4-13^{2} x\left(41\theta^4+82\theta^3+67\theta^2+26\theta+4\right)-2^{3} 13 x^{2}\left(471\theta^4+1788\theta^3+2555\theta^2+1534\theta+338\right)+2^{6} 13 x^{3}\left(251\theta^4+1014\theta^3+1798\theta^2+1413\theta+405\right)+2^{9} x^{4}\left(749\theta^4+436\theta^3-4908\theta^2-6266\theta-2145\right)-2^{12} x^{5}\left(379\theta^4+1270\theta^3+967\theta^2-42\theta-178\right)-2^{15} x^{6}\left(9\theta^4-156\theta^3-273\theta^2-156\theta-28\right)+2^{18} x^{7}\left(13\theta^4+26\theta^3+20\theta^2+7\theta+1\right)-2^{21} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 4, 68, 1552, 43156, ...
--> OEIS
Normalized instanton numbers (n0=1): 3, 226/13, 1621/13, 20666/13, 289056/13, ... ; Common denominator:...

Discriminant

\(-(z-1)(8z+1)(64z^2-48z+1)(-13+64z^2)^2\)

Local exponents

\(-\frac{ 1}{ 8}\sqrt{ 13}\)\(-\frac{ 1}{ 8}\)\(0\)\(\frac{ 3}{ 8}-\frac{ 1}{ 4}\sqrt{ 2}\)\(\frac{ 1}{ 8}\sqrt{ 13}\)\(\frac{ 3}{ 8}+\frac{ 1}{ 4}\sqrt{ 2}\)\(1\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)\(1\)
\(3\)\(1\)\(0\)\(1\)\(3\)\(1\)\(1\)\(1\)
\(4\)\(2\)\(0\)\(2\)\(4\)\(2\)\(2\)\(1\)

Note:

The operator has a second MUM-point at infinity, corresponding to operator 8.19.

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

62

New Number: 8.19 |  AESZ: 201  |  Superseeker: 32 7584  |  Hash: d21570c07bca6887061716b2d727fa75  

Degree: 8

\(\theta^4-2^{4} x\left(13\theta^4+26\theta^3+20\theta^2+7\theta+1\right)+2^{8} x^{2}\left(9\theta^4+192\theta^3+249\theta^2+114\theta+20\right)+2^{12} x^{3}\left(379\theta^4+246\theta^3-569\theta^2-318\theta-60\right)-2^{16} x^{4}\left(749\theta^4+2560\theta^3-1722\theta^2-1862\theta-474\right)-2^{20} 13 x^{5}\left(251\theta^4-10\theta^3+262\theta^2+145\theta+27\right)+2^{24} 13 x^{6}\left(471\theta^4+96\theta^3+17\theta^2+96\theta+42\right)+2^{28} 13^{2} x^{7}\left(41\theta^4+82\theta^3+67\theta^2+26\theta+4\right)-2^{35} 13^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 16, 752, 49408, 3805456, ...
--> OEIS
Normalized instanton numbers (n0=1): 32, -152, 7584, -160593, 7055200, ... ; Common denominator:...

Discriminant

\(-(128z-1)(16z+1)(256z^2-96z+1)(-1+3328z^2)^2\)

Local exponents

\(-\frac{ 1}{ 16}\)\(-\frac{ 1}{ 208}\sqrt{ 13}\)\(0\)\(\frac{ 1}{ 128}\)\(\frac{ 3}{ 16}-\frac{ 1}{ 8}\sqrt{ 2}\)\(\frac{ 1}{ 208}\sqrt{ 13}\)\(\frac{ 3}{ 16}+\frac{ 1}{ 8}\sqrt{ 2}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)\(1\)
\(1\)\(3\)\(0\)\(1\)\(1\)\(3\)\(1\)\(1\)
\(2\)\(4\)\(0\)\(2\)\(2\)\(4\)\(2\)\(1\)

Note:

This operator has a second MUM-point, corresponding to operator 8.18

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

63

New Number: 8.1 |  AESZ: 102  |  Superseeker: 8 1053  |  Hash: e928905653beb9d844e6a942f50d94ac  

Degree: 8

\(\theta^4-x(7\theta^2+7\theta+2)(11\theta^2+11\theta+3)-x^{2}\left(1049\theta^4+4100\theta^3+5689\theta^2+3178\theta+640\right)+2^{3} x^{3}\left(77\theta^4-462\theta^3-1420\theta^2-1053\theta-252\right)+2^{4} x^{4}\left(1041\theta^4+2082\theta^3-1406\theta^2-2447\theta-746\right)+2^{6} x^{5}\left(77\theta^4+770\theta^3+428\theta^2-93\theta-80\right)-2^{6} x^{6}\left(1049\theta^4+96\theta^3-317\theta^2+96\theta+100\right)-2^{9} x^{7}(7\theta^2+7\theta+2)(11\theta^2+11\theta+3)+2^{12} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 190, 8232, 432846, ...
--> OEIS
Normalized instanton numbers (n0=1): 8, 153/2, 1053, 49101/2, 670214, ... ; Common denominator:...

Discriminant

\((64z^2+88z-1)(z^2-11z-1)(-1+8z^2)^2\)

Local exponents

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

Note:

Hadamard product $a \ast b$. The operator has a second MUM-point at infinity with the same instanton numbers. In fact, there is a symmetry in the operator. It can be reduced to an operator with a single MUM point of degree 4, defined over $Q(\sqrt{2})$.

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

64

New Number: 8.22 |  AESZ: 284  |  Superseeker: 241/38 8729/19  |  Hash: dbe506beab1f66a0b331f15c91b7fcde  

Degree: 8

\(2^{2} 19^{2} \theta^4-2 19 x\left(3014\theta^4+5878\theta^3+4725\theta^2+1786\theta+266\right)+x^{2}\left(402002+1810054\theta+3057079\theta^2+2305502\theta^3+689717\theta^4\right)-x^{3}\left(1576582+6295992\theta+9142457\theta^2+5812350\theta^3+1438808\theta^4\right)+x^{4}\left(663471+3375833\theta+6297445\theta^2+5075392\theta^3+1395491\theta^4\right)+x^{5}\left(52928-604005\theta-2407768\theta^2-2657224\theta^3-834163\theta^4\right)-x^{6}\left(4832-148359\theta-572576\theta^2-692484\theta^3-277543\theta^4\right)-11 x^{7}\left(4625\theta^4+9100\theta^3+6395\theta^2+1845\theta+178\right)-11^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 7, 163, 5767, 247651, ...
--> OEIS
Normalized instanton numbers (n0=1): 241/38, 1353/38, 8729/19, 150334/19, 6399445/38, ... ; Common denominator:...

Discriminant

\(-(-1+78z-374z^2+425z^3+z^4)(38-25z+11z^2)^2\)

Local exponents

\(0\)\(\frac{ 25}{ 22}-\frac{ 1}{ 22}\sqrt{ 1047}I\)\(\frac{ 25}{ 22}+\frac{ 1}{ 22}\sqrt{ 1047}I\)\(#ND+#NDI\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(1\)
\(0\)\(1\)\(1\)\(1\)\(1\)
\(0\)\(3\)\(3\)\(1\)\(1\)
\(0\)\(4\)\(4\)\(2\)\(1\)

Note:

This operator has a second MUM-point at infinity, corresponding to operator 8.23

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

65

New Number: 8.23 |  AESZ: 285  |  Superseeker: -795/11 -1594688/11  |  Hash: 009974f32940428eb2d2d31380b138a9  

Degree: 8

\(11^{2} \theta^4+11 x\left(4625\theta^4+9400\theta^3+6845\theta^2+2145\theta+253\right)-x^{2}\left(4444+29513\theta+160382\theta^2+417688\theta^3+277543\theta^4\right)+x^{3}\left(834163\theta^4+679428\theta^3-558926\theta^2-423489\theta-72226\right)+x^{4}\left(94818+425155\theta+555785\theta^2-506572\theta^3-1395491\theta^4\right)+x^{5}\left(1438808\theta^4-57118\theta^3+338255\theta^2+307104\theta+49505\right)-x^{6}\left(33242+146466\theta+278875\theta^2+453366\theta^3+689717\theta^4\right)+2 19 x^{7}\left(3014\theta^4+6178\theta^3+5175\theta^2+2086\theta+341\right)-2^{2} 19^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -23, 3043, -620663, 154394851, ...
--> OEIS
Normalized instanton numbers (n0=1): -795/11, 89027/44, -1594688/11, 166273857/11, -21441641455/11, ... ; Common denominator:...

Discriminant

\(-(-1-425z+374z^2-78z^3+z^4)(11-25z+38z^2)^2\)

Local exponents

\(0\)\(\frac{ 25}{ 76}-\frac{ 1}{ 76}\sqrt{ 1047}I\)\(\frac{ 25}{ 76}+\frac{ 1}{ 76}\sqrt{ 1047}I\)\(#ND+#NDI\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(1\)
\(0\)\(1\)\(1\)\(1\)\(1\)
\(0\)\(3\)\(3\)\(1\)\(1\)
\(0\)\(4\)\(4\)\(2\)\(1\)

Note:

This operator has a second MUM-point at infinity, corresponding to operator 8.22

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

66

New Number: 8.26 |  AESZ: 301  |  Superseeker: 193/11 48570/11  |  Hash: a91db18876a9dfbf42b88f8d64c55d85  

Degree: 8

\(11^{2} \theta^4-11 x\left(1517\theta^4+3136\theta^3+2393\theta^2+825\theta+110\right)-x^{2}\left(24266+106953\theta+202166\theta^2+207620\theta^3+90362\theta^4\right)-x^{3}\left(53130+217437\theta+415082\theta^2+507996\theta^3+245714\theta^4\right)-x^{4}\left(15226+183269\theta+564786\theta^2+785972\theta^3+407863\theta^4\right)-x^{5}\left(25160+279826\theta+728323\theta^2+790148\theta^3+434831\theta^4\right)-2^{3} x^{6}\left(36361\theta^4+70281\theta^3+73343\theta^2+37947\theta+7644\right)-2^{4} 5 x^{7}\left(1307\theta^4+3430\theta^3+3877\theta^2+2162\theta+488\right)-2^{9} 5^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 10, 466, 32392, 2727826, ...
--> OEIS
Normalized instanton numbers (n0=1): 193/11, 1973/11, 48570/11, 1689283/11, 72444183/11, ... ; Common denominator:...

Discriminant

\(-(-1+143z+32z^2)(z+1)^2(20z^2+17z+11)^2\)

Local exponents

\(-\frac{ 143}{ 64}-\frac{ 19}{ 64}\sqrt{ 57}\)\(-1\)\(-\frac{ 17}{ 40}-\frac{ 1}{ 40}\sqrt{ 591}I\)\(-\frac{ 17}{ 40}+\frac{ 1}{ 40}\sqrt{ 591}I\)\(0\)\(-\frac{ 143}{ 64}+\frac{ 19}{ 64}\sqrt{ 57}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(0\)\(1\)\(1\)
\(1\)\(\frac{ 1}{ 2}\)\(3\)\(3\)\(0\)\(1\)\(1\)
\(2\)\(1\)\(4\)\(4\)\(0\)\(2\)\(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.27.

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

67

New Number: 8.27 |  AESZ: 302  |  Superseeker: 109/5 16777/5  |  Hash: e18ddbe4d66a3648b349130bcf119dc7  

Degree: 8

\(5^{2} \theta^4-5 x\left(1307\theta^4+1798\theta^3+1429\theta^2+530\theta+80\right)+2^{4} x^{2}\left(36361\theta^4+75163\theta^3+80666\theta^2+43340\theta+9120\right)-2^{6} x^{3}\left(434831\theta^4+949176\theta^3+966865\theta^2+545700\theta+118340\right)+2^{11} x^{4}\left(407863\theta^4+845480\theta^3+654048\theta^2+219839\theta+18634\right)-2^{16} x^{5}\left(245714\theta^4+474860\theta^3+365378\theta^2+71595\theta-11507\right)+2^{21} x^{6}\left(90362\theta^4+153828\theta^3+121478\theta^2+35967\theta+2221\right)-2^{26} 11 x^{7}\left(1517\theta^4+2932\theta^3+2087\theta^2+621\theta+59\right)-2^{31} 11^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 16, 664, 41920, 3350776, ...
--> OEIS
Normalized instanton numbers (n0=1): 109/5, 867/4, 16777/5, 662976/5, 26339071/5, ... ; Common denominator:...

Discriminant

\(-(-1+143z+32z^2)(32z-1)^2(2816z^2-136z+5)^2\)

Local exponents

\(-\frac{ 143}{ 64}-\frac{ 19}{ 64}\sqrt{ 57}\)\(0\)\(-\frac{ 143}{ 64}+\frac{ 19}{ 64}\sqrt{ 57}\)\(\frac{ 17}{ 704}-\frac{ 1}{ 704}\sqrt{ 591}I\)\(\frac{ 17}{ 704}+\frac{ 1}{ 704}\sqrt{ 591}I\)\(\frac{ 1}{ 32}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)
\(1\)\(0\)\(1\)\(3\)\(3\)\(\frac{ 1}{ 2}\)\(1\)
\(2\)\(0\)\(2\)\(4\)\(4\)\(1\)\(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.26

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

68

New Number: 8.28 |  AESZ: 303  |  Superseeker: 151/13 26293/13  |  Hash: e081c85684dd16a72eeaf5a1b139b912  

Degree: 8

\(13^{2} \theta^4-13 x\left(1505\theta^4+2746\theta^3+2127\theta^2+754\theta+104\right)+2^{2} x^{2}\left(22961\theta^4-2086\theta^3-55741\theta^2-41574\theta-9256\right)+2^{5} x^{3}\left(7524\theta^4+28098\theta^3+16131\theta^2+2691\theta-52\right)-2^{7} x^{4}\left(7241\theta^4+6214\theta^3+17522\theta^2+15423\theta+4146\right)-2^{8} x^{5}\left(6087\theta^4+1806\theta^3-3905\theta^2-3796\theta-1036\right)+2^{10} x^{6}\left(553\theta^4+4062\theta^3+4405\theta^2+1752\theta+220\right)+2^{14} x^{7}\left(82\theta^4+230\theta^3+275\theta^2+160\theta+37\right)+2^{18} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 8, 292, 15776, 1030036, ...
--> OEIS
Normalized instanton numbers (n0=1): 151/13, 1436/13, 26293/13, 719465/13, 24184128/13, ... ; Common denominator:...

Discriminant

\((z-1)(64z^3+304z^2+108z-1)(-13+44z+64z^2)^2\)

Local exponents

≈\(-4.362346\)\(-\frac{ 11}{ 32}-\frac{ 1}{ 32}\sqrt{ 329}\) ≈\(-0.396684\)\(0\) ≈\(0.009029\)\(-\frac{ 11}{ 32}+\frac{ 1}{ 32}\sqrt{ 329}\)\(1\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)
\(1\)\(3\)\(1\)\(0\)\(1\)\(3\)\(1\)\(1\)
\(2\)\(4\)\(2\)\(0\)\(2\)\(4\)\(2\)\(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.29

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

69

New Number: 8.29 |  AESZ: 304  |  Superseeker: -5 -641  |  Hash: cf055a245b1537ed4f2609fa56cf67aa  

Degree: 8

\(\theta^4+x\left(82\theta^4+98\theta^3+77\theta^2+28\theta+4\right)-x^{2}\left(636+2916\theta+4463\theta^2+1850\theta^3-553\theta^4\right)-2^{2} x^{3}\left(6087\theta^4+22542\theta^3+27199\theta^2+14916\theta+3136\right)-2^{5} x^{4}\left(7241\theta^4+22750\theta^3+42326\theta^2+29943\theta+7272\right)+2^{7} x^{5}\left(7524\theta^4+1998\theta^3-23019\theta^2-24627\theta-7186\right)+2^{8} x^{6}\left(22961\theta^4+93930\theta^3+88283\theta^2+28194\theta+1624\right)-2^{10} 13 x^{7}\left(1505\theta^4+3274\theta^3+2919\theta^2+1282\theta+236\right)+2^{14} 13^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -4, 112, -3712, 155536, ...
--> OEIS
Normalized instanton numbers (n0=1): -5, 469/8, -641, 50173/4, -276231, ... ; Common denominator:...

Discriminant

\((16z-1)(64z^3-432z^2-76z-1)(-1-11z+52z^2)^2\)

Local exponents

≈\(-0.157556\)\(\frac{ 11}{ 104}-\frac{ 1}{ 104}\sqrt{ 329}\) ≈\(-0.014327\)\(0\)\(\frac{ 1}{ 16}\)\(\frac{ 11}{ 104}+\frac{ 1}{ 104}\sqrt{ 329}\) ≈\(6.921883\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)
\(1\)\(3\)\(1\)\(0\)\(1\)\(3\)\(1\)\(1\)
\(2\)\(4\)\(2\)\(0\)\(2\)\(4\)\(2\)\(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.28

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

70

New Number: 8.2 |  AESZ: 104  |  Superseeker: 7 1271/3  |  Hash: d6bd0d1524954c8ce0a6421d295e9795  

Degree: 8

\(\theta^4-x(10\theta^2+10\theta+3)(7\theta^2+7\theta+2)-x^{2}\left(71\theta^4+1148\theta^3+1591\theta^2+886\theta+192\right)-2^{3} 3^{2} x^{3}\left(70\theta^4-420\theta^3-1289\theta^2-963\theta-240\right)-2^{4} 3^{2} x^{4}\left(143\theta^4+286\theta^3-1138\theta^2-1281\theta-414\right)+2^{6} 3^{4} x^{5}\left(70\theta^4+700\theta^3+391\theta^2-75\theta-76\right)-2^{6} 3^{4} x^{6}\left(71\theta^4-864\theta^3-1427\theta^2-864\theta-180\right)+2^{9} 3^{6} x^{7}(10\theta^2+10\theta+3)(7\theta^2+7\theta+2)+2^{12} 3^{8} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 150, 5208, 221094, ...
--> OEIS
Normalized instanton numbers (n0=1): 7, 93/2, 1271/3, 18507/2, 190710, ... ; Common denominator:...

Discriminant

\((9z+1)(8z-1)(72z-1)(z+1)(1+72z^2)^2\)

Local exponents

\(-1\)\(-\frac{ 1}{ 9}\)\(0-\frac{ 1}{ 12}\sqrt{ 2}I\)\(0\)\(0+\frac{ 1}{ 12}\sqrt{ 2}I\)\(\frac{ 1}{ 72}\)\(\frac{ 1}{ 8}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)
\(1\)\(1\)\(3\)\(0\)\(3\)\(1\)\(1\)\(1\)
\(2\)\(2\)\(4\)\(0\)\(4\)\(2\)\(2\)\(1\)

Note:

Hadamard product $a \ast c$. This operator has a second MUM-point at infinity with the same instanton point.
It is reducible to an operator of degree 4 with a single MUM-point defined over $Q(\sqrt{-2})$

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

71

New Number: 8.31 |  AESZ: 315  |  Superseeker: 38 26135  |  Hash: 44be55b95bb1c725c5aaa2c9a6635e89  

Degree: 8

\(5^{2} \theta^4-5^{2} x\left(239\theta^4+496\theta^3+368\theta^2+120\theta+15\right)-2 3 5 x^{2}\left(1727\theta^4+3206\theta^3+2341\theta^2+1090\theta+245\right)-3^{2} 5 x^{3}\left(1519\theta^4+7338\theta^3+14271\theta^2+8340\theta+1690\right)+3^{3} x^{4}\left(10358\theta^4-16622\theta^3-49763\theta^2-37900\theta-10210\right)+3^{4} 5 x^{5}\left(922\theta^4+3526\theta^3-1357\theta^2-3028\theta-1031\right)-3^{5} x^{6}\left(1219\theta^4-6030\theta^3-6441\theta^2-1740\theta+160\right)-2^{2} 3^{6} x^{7}\left(162\theta^4+234\theta^3+65\theta^2-52\theta-25\right)-2^{4} 3^{8} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 15, 1179, 140505, 20362059, ...
--> OEIS
Normalized instanton numbers (n0=1): 38, 3068/5, 26135, 7871998/5, 117518569, ... ; Common denominator:...

Discriminant

\(-(z+1)(81z^3+351z^2+246z-1)(-5-15z+36z^2)^2\)

Local exponents

≈\(-3.452681\)\(-1\) ≈\(-0.884694\)\(\frac{ 5}{ 24}-\frac{ 1}{ 24}\sqrt{ 105}\)\(0\) ≈\(0.004042\)\(\frac{ 5}{ 24}+\frac{ 1}{ 24}\sqrt{ 105}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)
\(1\)\(1\)\(1\)\(3\)\(0\)\(1\)\(3\)\(1\)
\(2\)\(2\)\(2\)\(4\)\(0\)\(2\)\(4\)\(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.32

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

72

New Number: 8.32 |  AESZ: 317  |  Superseeker: 69/4 14365/12  |  Hash: cda8cce31025f51636125bea67a820d1  

Degree: 8

\(2^{4} \theta^4-2^{2} 3 x\left(162\theta^4+414\theta^3+335\theta^2+128\theta+20\right)+3^{3} x^{2}\left(1219\theta^4+10906\theta^3+18963\theta^2+11824\theta+2708\right)+3^{5} 5 x^{3}\left(922\theta^4+162\theta^3-6403\theta^2-6576\theta-1964\right)-3^{7} x^{4}\left(10358\theta^4+58054\theta^3+62251\theta^2+29672\theta+4907\right)-3^{9} 5 x^{5}\left(1519\theta^4-1262\theta^3+1371\theta^2+4264\theta+1802\right)+2 3^{11} 5 x^{6}\left(1727\theta^4+3702\theta^3+3085\theta^2+882\theta+17\right)-3^{13} 5^{2} x^{7}\left(239\theta^4+460\theta^3+314\theta^2+84\theta+6\right)-3^{16} 5^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 15, 459, 19545, 1019259, ...
--> OEIS
Normalized instanton numbers (n0=1): 69/4, -30, 14365/12, 3015/2, 1376205/4, ... ; Common denominator:...

Discriminant

\(-(27z-1)(243z^3+2214z^2-117z+1)(-4-45z+405z^2)^2\)

Local exponents

≈\(-9.163702\)\(\frac{ 1}{ 18}-\frac{ 1}{ 90}\sqrt{ 105}\)\(0\) ≈\(0.010727\)\(\frac{ 1}{ 27}\) ≈\(0.041864\)\(\frac{ 1}{ 18}+\frac{ 1}{ 90}\sqrt{ 105}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)\(1\)
\(1\)\(3\)\(0\)\(1\)\(1\)\(1\)\(3\)\(1\)
\(2\)\(4\)\(0\)\(2\)\(2\)\(2\)\(4\)\(1\)

Note:

This operator has a second MUM-point at infininty corresponding to operator 8.31

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

73

New Number: 8.33 |  AESZ: 322  |  Superseeker: 4/3 95/3  |  Hash: a19da26bf1a7748e3b7e6151e803da30  

Degree: 8

\(3^{2} \theta^4+3 x\left(5\theta^4-122\theta^3-100\theta^2-39\theta-6\right)-x^{2}\left(5052+23736\theta+41729\theta^2+32600\theta^3+8603\theta^4\right)-2^{2} x^{3}\left(33304\theta^4+108297\theta^3+122347\theta^2+61470\theta+11712\right)-2^{2} x^{4}\left(180401\theta^4+547606\theta^3+638125\theta^2+339248\theta+69036\right)-2^{4} x^{5}\left(94934\theta^4+298745\theta^3+355667\theta^2+189660\theta+38224\right)-2^{4} x^{6}\left(73291\theta^4+204216\theta^3+190453\theta^2+68916\theta+6964\right)-2^{7} 3 x^{7}\left(811\theta^4+1886\theta^3+1804\theta^2+861\theta+174\right)-2^{10} 3^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 2, 46, 632, 16846, ...
--> OEIS
Normalized instanton numbers (n0=1): 4/3, 18, 95/3, 14575/12, 18158/3, ... ; Common denominator:...

Discriminant

\(-(-1+13z+827z^2+1928z^3+64z^4)(3+22z+12z^2)^2\)

Local exponents

\(-\frac{ 11}{ 12}-\frac{ 1}{ 12}\sqrt{ 85}\)\(-\frac{ 11}{ 12}+\frac{ 1}{ 12}\sqrt{ 85}\)\(0\)\(#ND+#NDI\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(0\)\(1\)\(1\)
\(3\)\(3\)\(0\)\(1\)\(1\)
\(4\)\(4\)\(0\)\(2\)\(1\)

Note:

This operator has a second MUM-point at infininty corresponding to operator 8.34

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

74

New Number: 8.34 |  AESZ: 323  |  Superseeker: 100/3 73111/3  |  Hash: 77c03b04c3a10350b5b0ccd2d204b18f  

Degree: 8

\(3^{2} \theta^4-3 x\left(811\theta^4+1358\theta^3+1012\theta^2+333\theta+42\right)-x^{2}\left(2424+7494\theta-17551\theta^2-88948\theta^3-73291\theta^4\right)-2^{3} x^{3}\left(94934\theta^4+80991\theta^3+29036\theta^2+5175\theta+420\right)+2^{4} x^{4}\left(180401\theta^4+173998\theta^3+77713\theta^2+15788\theta+708\right)-2^{7} x^{5}\left(33304\theta^4+24919\theta^3-2720\theta^2-8451\theta-2404\right)+2^{8} x^{6}\left(8603\theta^4+1812\theta^3-4453\theta^2-3666\theta-952\right)+2^{11} 3 x^{7}\left(5\theta^4+142\theta^3+296\theta^2+225\theta+60\right)-2^{14} 3^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 14, 1054, 120776, 16816846, ...
--> OEIS
Normalized instanton numbers (n0=1): 100/3, 1880/3, 73111/3, 4310384/3, 314245046/3, ... ; Common denominator:...

Discriminant

\(-(-1+241z-827z^2+104z^3+64z^4)(3-44z+48z^2)^2\)

Local exponents

\(0\)\(\frac{ 11}{ 24}-\frac{ 1}{ 24}\sqrt{ 85}\)\(\frac{ 11}{ 24}+\frac{ 1}{ 24}\sqrt{ 85}\)\(#ND+#NDI\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(1\)
\(0\)\(1\)\(1\)\(1\)\(1\)
\(0\)\(3\)\(3\)\(1\)\(1\)
\(0\)\(4\)\(4\)\(2\)\(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.33

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

75

New Number: 8.37 |  AESZ: 345  |  Superseeker: -12/11 357/11  |  Hash: 60f282ab4e1936cd96eb5ba12983db2d  

Degree: 8

\(11^{2} \theta^4+3 11 x\left(113\theta^4+184\theta^3+158\theta^2+66\theta+11\right)+2 x^{2}\left(28397\theta^4+95138\theta^3+128420\theta^2+77715\theta+17622\right)-3 x^{3}\left(3165\theta^4+180822\theta^3+560611\theta^2+539022\theta+167508\right)-3 x^{4}\left(233330\theta^4+1052614\theta^3+1424797\theta^2+774518\theta+145896\right)-3^{2} x^{5}\left(12866\theta^4-98902\theta^3-52127\theta^2+102028\theta+63723\right)+3^{2} x^{6}\left(183763\theta^4+473778\theta^3+427847\theta^2+147060\theta+11268\right)-2^{3} 3^{3} x^{7}\left(5006\theta^4+13414\theta^3+14935\theta^2+8228\theta+1869\right)+2^{6} 3^{7} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -3, 9, 141, -3879, ...
--> OEIS
Normalized instanton numbers (n0=1): -12/11, -28/11, 357/11, -1172/11, -5250/11, ... ; Common denominator:...

Discriminant

\((3z-1)(81z^3-457z^2-30z-1)(-11-21z+24z^2)^2\)

Local exponents

\(\frac{ 7}{ 16}-\frac{ 1}{ 48}\sqrt{ 1497}\) ≈\(-0.032637-0.033136I\) ≈\(-0.032637+0.033136I\)\(0\)\(\frac{ 1}{ 3}\)\(\frac{ 7}{ 16}+\frac{ 1}{ 48}\sqrt{ 1497}\) ≈\(5.707249\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)
\(3\)\(1\)\(1\)\(0\)\(1\)\(3\)\(1\)\(1\)
\(4\)\(2\)\(2\)\(0\)\(2\)\(4\)\(2\)\(1\)

Note:

This operator has a second MUM point at infinity corresponding to operator 8.38.

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

76

New Number: 8.38 |  AESZ: 346  |  Superseeker: 713/8 571555/2  |  Hash: 65579ed94e039ed095e1b5b7db3674ff  

Degree: 8

\(2^{6} \theta^4-2^{3} x\left(5006\theta^4+6610\theta^3+4729\theta^2+1424\theta+168\right)+3^{3} x^{2}\left(183763\theta^4+261274\theta^3+109091\theta^2+22352\theta+2040\right)-3^{7} x^{3}\left(12866\theta^4+150366\theta^3+321775\theta^2+141888\theta+21336\right)-3^{10} x^{4}\left(233330\theta^4-119294\theta^3-333065\theta^2-149446\theta-23109\right)-3^{14} x^{5}\left(3165\theta^4-168162\theta^3+37135\theta^2+52394\theta+11440\right)+2 3^{17} x^{6}\left(28397\theta^4+18450\theta^3+13388\theta^2+7299\theta+1586\right)+3^{22} 11 x^{7}\left(113\theta^4+268\theta^3+284\theta^2+150\theta+32\right)+3^{25} 11^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 21, 2889, 636357, 171536121, ...
--> OEIS
Normalized instanton numbers (n0=1): 713/8, 3274, 571555/2, 66913005/2, 20047292157/4, ... ; Common denominator:...

Discriminant

\((27z-1)(6561z^3+2430z^2+457z-1)(-8+567z+24057z^2)^2\)

Local exponents

≈\(-0.186267-0.189115I\) ≈\(-0.186267+0.189115I\)\(-\frac{ 7}{ 594}-\frac{ 1}{ 1782}\sqrt{ 1497}\)\(0\) ≈\(0.002163\)\(-\frac{ 7}{ 594}+\frac{ 1}{ 1782}\sqrt{ 1497}\)\(\frac{ 1}{ 27}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)
\(1\)\(1\)\(3\)\(0\)\(1\)\(3\)\(1\)\(1\)
\(2\)\(2\)\(4\)\(0\)\(2\)\(4\)\(2\)\(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.37

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

77

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

Degree: 8

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

Maple   LaTex

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
$Q(\sqrt{-1})$.

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

78

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

Degree: 8

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

Maple   LaTex

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 ...

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

79

New Number: 8.42 |  AESZ:  |  Superseeker: -4 140  |  Hash: 7bc3855c04953ca11620400320722844  

Degree: 8

\(\theta^4+2^{2} x\left(26\theta^4+34\theta^3+29\theta^2+12\theta+2\right)+2^{4} x^{2}\left(305\theta^4+662\theta^3+781\theta^2+436\theta+94\right)+2^{8} x^{3}\left(519\theta^4+1278\theta^3+1541\theta^2+933\theta+213\right)+2^{10} x^{4}\left(2266\theta^4+4988\theta^3+3535\theta^2+633\theta-162\right)+2^{14} 3 x^{5}\left(569\theta^4+1184\theta^3+740\theta^2-81\theta-128\right)+2^{18} 3 x^{6}\left(254\theta^4+354\theta^3+161\theta^2-33\theta-28\right)+2^{22} 3^{2} x^{7}\left(23\theta^4+34\theta^3+8\theta^2-9\theta-4\right)-2^{27} 3^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -8, 112, -1664, 23056, ...
--> OEIS
Normalized instanton numbers (n0=1): -4, 1/2, 140, 1025/2, -9196, ... ; Common denominator:...

Discriminant

\(-(32z+1)(1024z^3-896z^2-48z-1)(1+12z+192z^2)^2\)

Local exponents

\(-\frac{ 1}{ 32}-\frac{ 1}{ 96}\sqrt{ 39}I\)\(-\frac{ 1}{ 32}\)\(-\frac{ 1}{ 32}+\frac{ 1}{ 96}\sqrt{ 39}I\) ≈\(-0.025859-0.019623I\) ≈\(-0.025859+0.019623I\)\(0\) ≈\(0.926719\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)
\(3\)\(1\)\(3\)\(1\)\(1\)\(0\)\(1\)\(1\)
\(4\)\(2\)\(4\)\(2\)\(2\)\(0\)\(2\)\(1\)

Note:

This is operator "8.42" from ...

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

80

New Number: 8.43 |  AESZ:  |  Superseeker: 66/7 8716/7  |  Hash: 923554dba37f79c41bf0f67b875c36f7  

Degree: 8

\(7^{2} \theta^4-2 7 x\left(452\theta^4+640\theta^3+509\theta^2+189\theta+28\right)+2^{2} x^{2}\left(47156\theta^4+78224\theta^3+63963\theta^2+31010\theta+7000\right)-2^{5} x^{3}\left(77224\theta^4+150936\theta^3+155876\theta^2+86751\theta+19838\right)+2^{8} x^{4}\left(65988\theta^4+160584\theta^3+193653\theta^2+117501\theta+28198\right)-2^{12} x^{5}\left(15712\theta^4+46888\theta^3+63382\theta^2+41163\theta+10338\right)+2^{16} x^{6}\left(2088\theta^4+7272\theta^3+10589\theta^2+7140\theta+1828\right)-2^{22} x^{7}\left(36\theta^4+138\theta^3+206\theta^2+137\theta+34\right)+2^{26} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 8, 224, 10016, 547936, ...
--> OEIS
Normalized instanton numbers (n0=1): 66/7, 573/7, 8716/7, 197852/7, 5617614/7, ... ; Common denominator:...

Discriminant

\((1-96z+256z^2)(4z-1)^2(128z^2-88z+7)^2\)

Local exponents

\(0\)\(\frac{ 3}{ 16}-\frac{ 1}{ 8}\sqrt{ 2}\)\(\frac{ 11}{ 32}-\frac{ 1}{ 32}\sqrt{ 65}\)\(\frac{ 1}{ 4}\)\(\frac{ 3}{ 16}+\frac{ 1}{ 8}\sqrt{ 2}\)\(\frac{ 11}{ 32}+\frac{ 1}{ 32}\sqrt{ 65}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(1\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(3\)\(1\)
\(0\)\(2\)\(4\)\(1\)\(2\)\(4\)\(1\)

Note:

This is operator "8.43" from ...

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

81

New Number: 8.44 |  AESZ:  |  Superseeker: -64 -131904  |  Hash: 2d570a3dc1cbc5b6596272f33b48fc98  

Degree: 8

\(\theta^4-2^{5} x\left(36\theta^4+6\theta^3+8\theta^2+5\theta+1\right)+2^{8} x^{2}\left(2088\theta^4+1080\theta^3+1301\theta^2+574\theta+93\right)-2^{13} x^{3}\left(15712\theta^4+15960\theta^3+16990\theta^2+7785\theta+1381\right)+2^{18} x^{4}\left(65988\theta^4+103368\theta^3+107829\theta^2+52005\theta+9754\right)-2^{24} x^{5}\left(77224\theta^4+157960\theta^3+166412\theta^2+81089\theta+15251\right)+2^{30} x^{6}\left(47156\theta^4+110400\theta^3+112227\theta^2+50868\theta+8885\right)-2^{38} 7 x^{7}\left(452\theta^4+1168\theta^3+1301\theta^2+717\theta+160\right)+2^{46} 7^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 32, 2096, 172544, 15870736, ...
--> OEIS
Normalized instanton numbers (n0=1): -64, -2084, -131904, -10745878, -1015115456, ... ; Common denominator:...

Discriminant

\((1-192z+1024z^2)(128z-1)^2(14336z^2-352z+1)^2\)

Local exponents

\(0\)\(\frac{ 11}{ 896}-\frac{ 1}{ 896}\sqrt{ 65}\)\(\frac{ 3}{ 32}-\frac{ 1}{ 16}\sqrt{ 2}\)\(\frac{ 1}{ 128}\)\(\frac{ 11}{ 896}+\frac{ 1}{ 896}\sqrt{ 65}\)\(\frac{ 3}{ 32}+\frac{ 1}{ 16}\sqrt{ 2}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(3\)\(1\)\(\frac{ 1}{ 2}\)\(3\)\(1\)\(1\)
\(0\)\(4\)\(2\)\(1\)\(4\)\(2\)\(1\)

Note:

This is operator "8.44" from ...

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

82

New Number: 8.45 |  AESZ:  |  Superseeker: -12/5 -20  |  Hash: 52e4f6959f297529016ddef66a399c12  

Degree: 8

\(5^{2} \theta^4+2^{2} 5 x\left(19\theta^4+86\theta^3+73\theta^2+30\theta+5\right)+2^{4} x^{2}\left(709\theta^4+4252\theta^3+7339\theta^2+4830\theta+1165\right)-2^{8} x^{3}\left(420\theta^4+114\theta^3-3294\theta^2-3960\theta-1325\right)-2^{10} x^{4}\left(949\theta^4+6782\theta^3+11350\theta^2+7719\theta+1889\right)+2^{12} x^{5}\left(1315\theta^4+4282\theta^3+7199\theta^2+5744\theta+1691\right)+2^{14} x^{6}\left(613\theta^4+1560\theta^3+973\theta^2-216\theta-249\right)+2^{18} x^{7}\left(11\theta^4-2\theta^3-40\theta^2-39\theta-11\right)-2^{20} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -4, -4, 464, -4244, ...
--> OEIS
Normalized instanton numbers (n0=1): -12/5, -83/5, -20, 3941/20, -13872/5, ... ; Common denominator:...

Discriminant

\(-(8z+1)(128z^3-624z^2-20z-1)(-5+32z+32z^2)^2\)

Local exponents

\(-\frac{ 1}{ 2}-\frac{ 1}{ 8}\sqrt{ 26}\)\(-\frac{ 1}{ 8}\) ≈\(-0.016083-0.036516I\) ≈\(-0.016083+0.036516I\)\(0\)\(-\frac{ 1}{ 2}+\frac{ 1}{ 8}\sqrt{ 26}\) ≈\(4.907166\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)
\(3\)\(1\)\(1\)\(1\)\(0\)\(3\)\(1\)\(1\)
\(4\)\(2\)\(2\)\(2\)\(0\)\(4\)\(2\)\(1\)

Note:

This is operator "8.45" from ...

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

83

New Number: 8.46 |  AESZ:  |  Superseeker: 192 616896  |  Hash: e38642e59fc2f9e3117437fcdbfe450e  

Degree: 8

\(\theta^4-2^{5} x\left(11\theta^4+46\theta^3+32\theta^2+9\theta+1\right)-2^{8} x^{2}\left(613\theta^4+892\theta^3-29\theta^2-66\theta-7\right)-2^{13} x^{3}\left(1315\theta^4+978\theta^3+2243\theta^2+1068\theta+179\right)+2^{18} x^{4}\left(949\theta^4-2986\theta^3-3302\theta^2-1569\theta-313\right)+2^{23} x^{5}\left(420\theta^4+1566\theta^3-1116\theta^2-1290\theta-353\right)-2^{26} x^{6}\left(709\theta^4-1416\theta^3-1163\theta^2-72\theta+131\right)-2^{31} 5 x^{7}\left(19\theta^4-10\theta^3-71\theta^2-66\theta-19\right)-2^{36} 5^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 32, 6224, 1859072, 679223056, ...
--> OEIS
Normalized instanton numbers (n0=1): 192, 3108, 616896, 73692781, 15330708544, ... ; Common denominator:...

Discriminant

\(-(16z+1)(16384z^3+2560z^2+624z-1)(-1-128z+2560z^2)^2\)

Local exponents

≈\(-0.078921-0.179189I\) ≈\(-0.078921+0.179189I\)\(-\frac{ 1}{ 16}\)\(\frac{ 1}{ 40}-\frac{ 1}{ 160}\sqrt{ 26}\)\(0\) ≈\(0.001592\)\(\frac{ 1}{ 40}+\frac{ 1}{ 160}\sqrt{ 26}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)
\(1\)\(1\)\(1\)\(3\)\(0\)\(1\)\(3\)\(1\)
\(2\)\(2\)\(2\)\(4\)\(0\)\(2\)\(4\)\(1\)

Note:

This is operator "8.46" from ...

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

84

New Number: 8.4 |  AESZ: 160  |  Superseeker: 6 -325  |  Hash: 8ce8667fe6e49ce6625fafe044b1641b  

Degree: 8

\(\theta^4-3 x(3\theta^2+3\theta+1)(7\theta^2+7\theta+2)+3^{2} x^{2}\left(171\theta^4+396\theta^3+555\theta^2+318\theta+64\right)-2^{3} 3^{4} x^{3}\left(21\theta^4-126\theta^3-386\theta^2-291\theta-76\right)+2^{4} 3^{5} x^{4}\left(147\theta^4+294\theta^3+102\theta^2-45\theta-14\right)+2^{6} 3^{7} x^{5}\left(21\theta^4+210\theta^3+118\theta^2-19\theta-24\right)+2^{6} 3^{8} x^{6}\left(171\theta^4+288\theta^3+393\theta^2+288\theta+76\right)+2^{9} 3^{10} x^{7}(3\theta^2+3\theta+1)(7\theta^2+7\theta+2)+2^{12} 3^{12} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 90, 1176, 3114, ...
--> OEIS
Normalized instanton numbers (n0=1): 6, 6, -325, -1977/2, -5421, ... ; Common denominator:...

Discriminant

\((27z^2+9z+1)(1728z^2-72z+1)(1+216z^2)^2\)

Local exponents

\(-\frac{ 1}{ 6}-\frac{ 1}{ 18}\sqrt{ 3}I\)\(-\frac{ 1}{ 6}+\frac{ 1}{ 18}\sqrt{ 3}I\)\(0-\frac{ 1}{ 36}\sqrt{ 6}I\)\(0\)\(0+\frac{ 1}{ 36}\sqrt{ 6}I\)\(\frac{ 1}{ 48}-\frac{ 1}{ 144}\sqrt{ 3}I\)\(\frac{ 1}{ 48}+\frac{ 1}{ 144}\sqrt{ 3}I\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)
\(1\)\(1\)\(3\)\(0\)\(3\)\(1\)\(1\)\(1\)
\(2\)\(2\)\(4\)\(0\)\(4\)\(2\)\(2\)\(1\)

Note:

Hadamard product $a \ast f$. 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$Q(\sqrt{6})$.

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

85

New Number: 8.52 |  AESZ:  |  Superseeker: 416 9734432  |  Hash: 87729f275f24cb2daf88133571476576  

Degree: 8

\(\theta^4+2^{4} x\left(176\theta^4-32\theta^3+4\theta^2+20\theta+5\right)+2^{12} x^{2}\left(640\theta^4+256\theta^3+680\theta^2+224\theta+27\right)+2^{22} x^{3}\left(220\theta^4+648\theta^3+596\theta^2+348\theta+85\right)+2^{30} x^{4}\left(116\theta^4+1024\theta^3+1608\theta^2+1072\theta+281\right)-2^{38} x^{5}\left(32\theta^4-448\theta^3-1588\theta^2-1404\theta-437\right)-2^{46} x^{6}\left(80\theta^4+288\theta^3-88\theta^2-384\theta-179\right)-2^{57} x^{7}\left(2\theta^4+28\theta^3+56\theta^2+42\theta+11\right)+2^{66} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -80, 6928, -597248, 95243536, ...
--> OEIS
Normalized instanton numbers (n0=1): 416, -52752, 9734432, -2404009688, 687625871328, ... ; Common denominator:...

Discriminant

\((1+256z+65536z^2)(256z+1)^2(131072z^2-1024z-1)^2\)

Local exponents

\(-\frac{ 1}{ 256}\)\(-\frac{ 1}{ 512}-\frac{ 1}{ 512}\sqrt{ 3}I\)\(-\frac{ 1}{ 512}+\frac{ 1}{ 512}\sqrt{ 3}I\)\(\frac{ 1}{ 256}-\frac{ 1}{ 512}\sqrt{ 6}\)\(0\)\(\frac{ 1}{ 256}+\frac{ 1}{ 512}\sqrt{ 6}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)
\(\frac{ 1}{ 2}\)\(1\)\(1\)\(3\)\(0\)\(3\)\(1\)
\(1\)\(2\)\(2\)\(4\)\(0\)\(4\)\(1\)

Note:

This is operator "8.52" from ...

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

86

New Number: 8.53 |  AESZ:  |  Superseeker: -64 -64320  |  Hash: 0714a480e1c587ebada71771f7e3b555  

Degree: 8

\(\theta^4-2^{5} x\left(2\theta^4-20\theta^3-16\theta^2-6\theta-1\right)-2^{8} x^{2}\left(80\theta^4+32\theta^3-472\theta^2-336\theta-91\right)-2^{14} x^{3}\left(32\theta^4+576\theta^3-52\theta^2-300\theta-141\right)+2^{20} x^{4}\left(116\theta^4-560\theta^3-768\theta^2-464\theta-91\right)+2^{26} x^{5}\left(220\theta^4+232\theta^3-28\theta^2-220\theta-95\right)+2^{30} x^{6}\left(640\theta^4+2304\theta^3+3752\theta^2+2928\theta+867\right)+2^{36} x^{7}\left(176\theta^4+736\theta^3+1156\theta^2+788\theta+197\right)+2^{46} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -32, 1168, -43520, 1777936, ...
--> OEIS
Normalized instanton numbers (n0=1): -64, -1604, -64320, -3255802, -191614656, ... ; Common denominator:...

Discriminant

\((1+64z+4096z^2)(64z+1)^2(2048z^2+128z-1)^2\)

Local exponents

\(-\frac{ 1}{ 32}-\frac{ 1}{ 64}\sqrt{ 6}\)\(-\frac{ 1}{ 64}\)\(-\frac{ 1}{ 128}-\frac{ 1}{ 128}\sqrt{ 3}I\)\(-\frac{ 1}{ 128}+\frac{ 1}{ 128}\sqrt{ 3}I\)\(0\)\(-\frac{ 1}{ 32}+\frac{ 1}{ 64}\sqrt{ 6}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(0\)\(1\)\(1\)
\(3\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(0\)\(3\)\(1\)
\(4\)\(1\)\(2\)\(2\)\(0\)\(4\)\(1\)

Note:

This is operator "8.53" from ...

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

87

New Number: 8.56 |  AESZ:  |  Superseeker: 80 266256  |  Hash: b561c9f1501dce5c055c95391a2176d3  

Degree: 8

\(\theta^4-2^{4} x\left(34\theta^4+44\theta^3+31\theta^2+9\theta+1\right)+2^{9} x^{2}\left(94\theta^4-14\theta^3-168\theta^2-98\theta-19\right)-2^{12} x^{3}\left(368\theta^4-1104\theta^3-1505\theta^2-549\theta-60\right)+2^{16} x^{4}\left(28\theta^4-2740\theta^3-154\theta^2+928\theta+331\right)+2^{20} x^{5}\left(678\theta^4+1116\theta^3-2997\theta^2-2295\theta-505\right)-2^{26} x^{6}\left(94\theta^4-561\theta^3-508\theta^2-132\theta+6\right)-2^{28} 5 x^{7}\left(92\theta^4+160\theta^3+97\theta^2+17\theta-2\right)-2^{32} 5^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 16, 2512, 533248, 138259216, ...
--> OEIS
Normalized instanton numbers (n0=1): 80, 3554, 266256, 31532007, 4663446128, ... ; Common denominator:...

Discriminant

\(-(16z+1)(4096z^3+4864z^2+432z-1)(1-64z+1280z^2)^2\)

Local exponents

≈\(-1.090586\) ≈\(-0.099171\)\(-\frac{ 1}{ 16}\)\(0\) ≈\(0.002257\)\(\frac{ 1}{ 40}-\frac{ 1}{ 80}I\)\(\frac{ 1}{ 40}+\frac{ 1}{ 80}I\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(3\)\(3\)\(1\)
\(2\)\(2\)\(2\)\(0\)\(2\)\(4\)\(4\)\(1\)

Note:

This is operator "8.56" from ...

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

88

New Number: 8.57 |  AESZ:  |  Superseeker: -36/5 -380  |  Hash: c2a931d298755811a60b7f8e5dd3afbe  

Degree: 8

\(5^{2} \theta^4+2^{2} 5 x\left(92\theta^4+208\theta^3+169\theta^2+65\theta+10\right)+2^{6} x^{2}\left(94\theta^4+937\theta^3+1739\theta^2+1175\theta+285\right)-2^{6} x^{3}\left(678\theta^4+1596\theta^3-2277\theta^2-4335\theta-1645\right)-2^{8} x^{4}\left(28\theta^4+2852\theta^3+8234\theta^2+7096\theta+2017\right)+2^{10} x^{5}\left(368\theta^4+2576\theta^3+4015\theta^2+2323\theta+456\right)-2^{13} x^{6}\left(94\theta^4+390\theta^3+438\theta^2+180\theta+19\right)+2^{14} x^{7}\left(34\theta^4+92\theta^3+103\theta^2+57\theta+13\right)-2^{16} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -8, 172, -5696, 231916, ...
--> OEIS
Normalized instanton numbers (n0=1): -36/5, 132/5, -380, 112043/20, -560656/5, ... ; Common denominator:...

Discriminant

\(-(4z+1)(64z^3-432z^2-76z-1)(5-16z+16z^2)^2\)

Local exponents

\(-\frac{ 1}{ 4}\) ≈\(-0.157556\) ≈\(-0.014327\)\(0\)\(\frac{ 1}{ 2}-\frac{ 1}{ 4}I\)\(\frac{ 1}{ 2}+\frac{ 1}{ 4}I\) ≈\(6.921883\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(1\)
\(1\)\(1\)\(1\)\(0\)\(3\)\(3\)\(1\)\(1\)
\(2\)\(2\)\(2\)\(0\)\(4\)\(4\)\(2\)\(1\)

Note:

This is operator "8.57" from ...

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

89

New Number: 8.58 |  AESZ:  |  Superseeker: 286 12179050/3  |  Hash: 870f2e78b48eb5ee8f5de2f6a438f2b8  

Degree: 8

\(\theta^4-x\left(1114\theta^4+2444\theta^3+1704\theta^2+482\theta+51\right)-x^{2}\left(85922\theta^4+94748\theta^3-21782\theta^2-21164\theta-3273\right)-3^{2} x^{3}\left(173242\theta^4+41004\theta^3+55912\theta^2+32322\theta+5679\right)+3^{2} x^{4}\left(189512\theta^4-918380\theta^3-841954\theta^2-306732\theta-47331\right)+3^{4} x^{5}\left(30338\theta^4+90716\theta^3-87560\theta^2-90566\theta-23193\right)-3^{4} x^{6}\left(19406\theta^4-68364\theta^3-62162\theta^2-14148\theta+1989\right)-3^{6} 5 x^{7}\left(278\theta^4+340\theta^3+8\theta^2-162\theta-63\right)-3^{8} 5^{2} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 51, 18267, 10280301, 7092708939, ...
--> OEIS
Normalized instanton numbers (n0=1): 286, 38919/2, 12179050/3, 2393489451/2, 439227114444, ... ; Common denominator:...

Discriminant

\(-(z+1)(81z^3+549z^2+1187z-1)(-1-36z+45z^2)^2\)

Local exponents

≈\(-3.38931-1.781181I\) ≈\(-3.38931+1.781181I\)\(-1\)\(\frac{ 2}{ 5}-\frac{ 1}{ 15}\sqrt{ 41}\)\(0\) ≈\(0.000842\)\(\frac{ 2}{ 5}+\frac{ 1}{ 15}\sqrt{ 41}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)
\(1\)\(1\)\(1\)\(3\)\(0\)\(1\)\(3\)\(1\)
\(2\)\(2\)\(2\)\(4\)\(0\)\(2\)\(4\)\(1\)

Note:

This is operator "8.58" from ...

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  

90

New Number: 8.59 |  AESZ:  |  Superseeker: -26/5 -234/5  |  Hash: 53885e46a1519d98ee4697de1c109214  

Degree: 8

\(5^{2} \theta^4+5 x\left(278\theta^4+772\theta^3+656\theta^2+270\theta+45\right)+x^{2}\left(19406\theta^4+145988\theta^3+259366\theta^2+172540\theta+41745\right)-3^{2} x^{3}\left(30338\theta^4+30636\theta^3-177680\theta^2-235350\theta-80565\right)-3^{2} x^{4}\left(189512\theta^4+1676428\theta^3+3050258\theta^2+2136012\theta+525339\right)+3^{4} x^{5}\left(173242\theta^4+651964\theta^3+972352\theta^2+649458\theta+161507\right)+3^{4} x^{6}\left(85922\theta^4+248940\theta^3+209506\theta^2+37044\theta-12717\right)+3^{6} x^{7}\left(1114\theta^4+2012\theta^3+1056\theta^2+50\theta-57\right)-3^{8} x^{8}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -9, 123, -1719, 17739, ...
--> OEIS
Normalized instanton numbers (n0=1): -26/5, -177/10, -234/5, -1837/2, -27716/5, ... ; Common denominator:...

Discriminant

\(-(9z+1)(9z^3-1187z^2-61z-1)(-5+36z+9z^2)^2\)

Local exponents

\(-2-\frac{ 1}{ 3}\sqrt{ 41}\)\(-\frac{ 1}{ 9}\) ≈\(-0.025688-0.0135I\) ≈\(-0.025688+0.0135I\)\(0\)\(-2+\frac{ 1}{ 3}\sqrt{ 41}\) ≈\(131.940265\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)
\(3\)\(1\)\(1\)\(1\)\(0\)\(3\)\(1\)\(1\)
\(4\)\(2\)\(2\)\(2\)\(0\)\(4\)\(2\)\(1\)

Note:

This is operator "8.59" from ...

Show more...  or download as   plain text  |  PDF  |  Maple  |  LaTex  


 1-30  31-60  61-90  91-113