Summary

You searched for: Spectrum0=0,1/2,3/2,2

Your search produced 17 matches

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1

New Number: 4.35 |  AESZ:  |  Superseeker: -16 -1744  |  Hash: cd392ce4c33f242f5d17e59976d0ea4f  

Degree: 4

\(\theta^4-2^{4} x\left(23\theta^4+14\theta^3+13\theta^2+6\theta+1\right)+2^{11} x^{2}\theta(21\theta^3+24\theta^2+18\theta+4)-2^{16} x^{3}(2\theta+1)(10\theta^3+7\theta^2-5\theta-4)-2^{23} x^{4}(2\theta+1)(\theta+1)^2(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 16, 912, 67840, 5839120, ...
--> OEIS
Normalized instanton numbers (n0=1): -16, -106, -1744, -29526, -644016, ... ; Common denominator:...

Discriminant

\(-(16z+1)(128z-1)^3\)

Local exponents

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

Note:

Operator equivalent to 3.34, equivalent to
AESZ 107 $=d \ast d$.

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2

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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3

New Number: 4.71 |  AESZ: 353  |  Superseeker: -4 -1580/9  |  Hash: 33845d8200fe810109063e352fbfc8b1  

Degree: 4

\(\theta^4-2^{2} x\left(52\theta^4+40\theta^3+37\theta^2+17\theta+3\right)+2^{4} x^{2}\left(960\theta^4+1536\theta^3+1512\theta^2+688\theta+123\right)-2^{8} x^{3}\left(1792\theta^4+4608\theta^3+5184\theta^2+2816\theta+597\right)+2^{14} x^{4}(4\theta+5)^2(4\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 324, 11856, 504900, ...
--> OEIS
Normalized instanton numbers (n0=1): -4, -24, -1580/9, -1580, -17120, ... ; Common denominator:...

Discriminant

\((16z-1)(64z-1)^3\)

Local exponents

\(0\)\(\frac{ 1}{ 64}\)\(\frac{ 1}{ 16}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 3}{ 4}\)
\(0\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 3}{ 4}\)
\(0\)\(\frac{ 3}{ 2}\)\(1\)\(\frac{ 5}{ 4}\)
\(0\)\(2\)\(2\)\(\frac{ 5}{ 4}\)

Note:

Sporadic Operator, reducible to 3.33, so not a primary operator.

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4

New Number: 12.5 |  AESZ:  |  Superseeker: 4 2252/9  |  Hash: bb257a283455fdd1fa17fef9649505e3  

Degree: 12

\(\theta^4+2^{2} x\left(43\theta^4+22\theta^3+25\theta^2+14\theta+3\right)+2^{4} x^{2}\left(753\theta^4+924\theta^3+1107\theta^2+622\theta+141\right)+2^{7} x^{3}\left(3377\theta^4+7218\theta^3+9261\theta^2+5764\theta+1455\right)+2^{10} x^{4}\left(7570\theta^4+24718\theta^3+34375\theta^2+21933\theta+5310\right)+2^{12} 3^{2} x^{5}\left(901\theta^4+5118\theta^3+5777\theta^2-84\theta-1829\right)-2^{14} 3^{2} x^{6}\left(7783\theta^4+33872\theta^3+83851\theta^2+107556\theta+49489\right)-2^{17} 3^{3} x^{7}\left(4895\theta^4+28154\theta^3+69267\theta^2+83564\theta+36929\right)-2^{20} 3^{4} x^{8}\left(44\theta^4+528\theta^3+247\theta^2+240\theta+274\right)+2^{23} 3^{5} x^{9}\left(664\theta^4+4760\theta^3+13781\theta^2+17353\theta+7679\right)+2^{26} 3^{6} x^{10}(\theta+1)(109\theta^3+651\theta^2+1373\theta+933)-2^{29} 3^{7} x^{11}(\theta+1)(\theta+2)(27\theta^2+153\theta+199)-2^{33} 3^{9} x^{12}(\theta+1)(\theta+2)^2(\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -12, 180, -2736, 42948, ...
--> OEIS
Normalized instanton numbers (n0=1): 4, -31, 2252/9, -11109/4, 33312, ... ; Common denominator:...

Discriminant

\(-(16z+1)(432z^2+36z+1)(24z+1)^2(288z^2+48z+1)^2(8z-1)^3\)

Local exponents

\(-\frac{ 1}{ 12}-\frac{ 1}{ 24}\sqrt{ 2}\)\(-\frac{ 1}{ 16}\)\(-\frac{ 1}{ 24}-\frac{ 1}{ 72}\sqrt{ 3}I\)\(-\frac{ 1}{ 24}\)\(-\frac{ 1}{ 24}+\frac{ 1}{ 72}\sqrt{ 3}I\)\(-\frac{ 1}{ 12}+\frac{ 1}{ 24}\sqrt{ 2}\)\(0\)\(\frac{ 1}{ 8}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(0\)\(\frac{ 1}{ 2}\)\(2\)
\(3\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(3\)\(0\)\(\frac{ 3}{ 2}\)\(2\)
\(4\)\(2\)\(2\)\(1\)\(2\)\(4\)\(0\)\(2\)\(3\)

Note:

This is operator "12.5" from ...

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5

New Number: 13.10 |  AESZ:  |  Superseeker: 4 -628/9  |  Hash: 2a9fda379889eb2fd218bd01f2520f7a  

Degree: 13

\(\theta^4-2^{2} x\left(35\theta^4+38\theta^3+35\theta^2+16\theta+3\right)+2^{4} x^{2}\left(546\theta^4+1068\theta^3+1287\theta^2+790\theta+201\right)-2^{6} x^{3}\left(4928\theta^4+12888\theta^3+17829\theta^2+12673\theta+3693\right)+2^{8} x^{4}\left(28123\theta^4+88408\theta^3+131977\theta^2+98226\theta+29511\right)-2^{10} 3^{2} x^{5}\left(11315\theta^4+41094\theta^3+65088\theta^2+47691\theta+13532\right)+2^{13} 3^{2} x^{6}\left(11674\theta^4+48674\theta^3+79399\theta^2+52683\theta+11716\right)-2^{15} 3^{3} x^{7}\left(2063\theta^4+11102\theta^3+11184\theta^2-9217\theta-10762\right)-2^{17} 3^{4} x^{8}\left(3277\theta^4+16284\theta^3+42329\theta^2+57018\theta+27266\right)+2^{20} 3^{5} x^{9}\left(1124\theta^4+7114\theta^3+18121\theta^2+22265\theta+10018\right)+2^{24} 3^{6} x^{10}(\theta+1)(\theta^3-105\theta^2-277\theta-267)-2^{25} 3^{7} x^{11}(\theta+1)(\theta+2)(93\theta^2+441\theta+607)+2^{27} 3^{10} x^{12}(\theta+3)(\theta+2)(\theta+1)(\theta+6)+2^{30} 3^{10} x^{13}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 180, 2928, 47556, ...
--> OEIS
Normalized instanton numbers (n0=1): 4, 5, -628/9, -2823/4, 672, ... ; Common denominator:...

Discriminant

\((8z-1)(10368z^3-1728z^2+72z-1)(12z-1)^2(288z^2-24z+1)^2(4z+1)^3\)

Local exponents

\(-\frac{ 1}{ 4}\)\(0\) ≈\(0.027033-0.011216I\) ≈\(0.027033+0.011216I\)\(\frac{ 1}{ 24}-\frac{ 1}{ 24}I\)\(\frac{ 1}{ 24}+\frac{ 1}{ 24}I\)\(\frac{ 1}{ 12}\) ≈\(0.112601\)\(\frac{ 1}{ 8}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(\frac{ 1}{ 2}\)\(0\)\(1\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(2\)
\(\frac{ 3}{ 2}\)\(0\)\(1\)\(1\)\(3\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(3\)
\(2\)\(0\)\(2\)\(2\)\(4\)\(4\)\(1\)\(2\)\(2\)\(4\)

Note:

This is operator "13.10" from ...

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6

New Number: 13.11 |  AESZ:  |  Superseeker: 7 -2044/9  |  Hash: d6e183df7853fe5068c8b8cdeb3f63cb  

Degree: 13

\(\theta^4-x\left(98\theta^4+164\theta^3+137\theta^2+55\theta+9\right)+x^{2}\left(3822\theta^4+11400\theta^3+14901\theta^2+8746\theta+2007\right)-x^{3}\left(64148\theta^4+196344\theta^3+271665\theta^2+199855\theta+60354\right)+x^{4}\left(802771\theta^4+2242504\theta^3+2203855\theta^2+1316868\theta+390636\right)-2 3 x^{5}\left(1040145\theta^4+2982426\theta^3+3578912\theta^2+1897395\theta+345411\right)+2 3^{2} x^{6}\left(1927994\theta^4+4917832\theta^3+7329041\theta^2+5154630\theta+1338003\right)-2 3^{5} x^{7}\left(219316\theta^4+761432\theta^3+1064075\theta^2+703129\theta+181966\right)+3^{4} x^{8}\left(754759\theta^4+7471824\theta^3+13904030\theta^2+8830464\theta+1544112\right)+3^{7} x^{9}\left(174966\theta^4+736236\theta^3+1307237\theta^2+1340471\theta+568265\right)-3^{10} x^{10}(\theta+1)(8018\theta^3+62342\theta^2+139257\theta+108861)-3^{9} x^{11}(\theta+1)(\theta+2)(28988\theta^2+81396\theta+36331)+3^{12} x^{12}(\theta+3)(\theta+2)(\theta+1)(1061\theta+5386)+2 3^{15} 17 x^{13}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 9, 135, 2115, 18063, ...
--> OEIS
Normalized instanton numbers (n0=1): 7, -31/4, -2044/9, -1380, -8520, ... ; Common denominator:...

Discriminant

\((2z-1)(4131z^3-2187z^2+81z-1)(3z-1)^2(81z^2-6z+1)^2(z+1)^3\)

Local exponents

\(-1\)\(0\) ≈\(0.019487-0.01067I\) ≈\(0.019487+0.01067I\)\(\frac{ 1}{ 27}-\frac{ 2}{ 27}\sqrt{ 2}I\)\(\frac{ 1}{ 27}+\frac{ 2}{ 27}\sqrt{ 2}I\)\(\frac{ 1}{ 3}\) ≈\(0.490438\)\(\frac{ 1}{ 2}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(\frac{ 1}{ 2}\)\(0\)\(1\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(2\)
\(\frac{ 3}{ 2}\)\(0\)\(1\)\(1\)\(3\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(3\)
\(2\)\(0\)\(2\)\(2\)\(4\)\(4\)\(1\)\(2\)\(2\)\(4\)

Note:

This is operator "13.11" from ...

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7

New Number: 13.6 |  AESZ:  |  Superseeker: 2 421/9  |  Hash: 679aa37a05aafe03e8d68785d566fcfb  

Degree: 13

\(\theta^4-x\left(217\theta^4+178\theta^3+178\theta^2+89\theta+18\right)+x^{2}\left(6192+24334\theta+39795\theta^2+33324\theta^3+20643\theta^4\right)-2^{3} x^{3}\left(139307\theta^4+333558\theta^3+457560\theta^2+315505\theta+89244\right)+2^{4} x^{4}\left(2283535\theta^4+7259062\theta^3+11103058\theta^2+8192571\theta+2419362\right)-2^{6} 3 x^{5}\left(3630237\theta^4+14551206\theta^3+23954402\theta^2+17624013\theta+4953960\right)+2^{6} 3^{2} x^{6}\left(9379387\theta^4+48172928\theta^3+74157721\theta^2+31932048\theta-1833876\right)+2^{9} 3^{5} x^{7}\left(495945\theta^4+2307886\theta^3+6892788\theta^2+10676039\theta+5452406\right)-2^{12} 3^{4} x^{8}\left(5269994\theta^4+31826568\theta^3+83327461\theta^2+106595346\theta+49104855\right)+2^{15} 3^{7} x^{9}\left(129774\theta^4+976140\theta^3+2673571\theta^2+3442327\theta+1597000\right)+2^{18} 3^{10} x^{10}(\theta+1)(6759\theta^3+40481\theta^2+97855\theta+79397)-2^{21} 3^{9} x^{11}(\theta+1)(\theta+2)(29107\theta^2+160713\theta+251822)-2^{27} 3^{12} x^{12}(\theta+3)(\theta+2)(\theta+1)(17\theta+4)+2^{29} 3^{15} 5 x^{13}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 378, 8280, 187434, ...
--> OEIS
Normalized instanton numbers (n0=1): 2, -3/2, 421/9, -519/2, 285, ... ; Common denominator:...

Discriminant

\((16z-1)(19440z^3-2187z^2+81z-1)(24z-1)^2(648z^2-48z+1)^2(8z+1)^3\)

Local exponents

\(-\frac{ 1}{ 8}\)\(0\) ≈\(0.032165-0.005771I\) ≈\(0.032165+0.005771I\)\(\frac{ 1}{ 27}-\frac{ 1}{ 108}\sqrt{ 2}I\)\(\frac{ 1}{ 27}+\frac{ 1}{ 108}\sqrt{ 2}I\)\(\frac{ 1}{ 24}\) ≈\(0.04817\)\(\frac{ 1}{ 16}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(\frac{ 1}{ 2}\)\(0\)\(1\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(2\)
\(\frac{ 3}{ 2}\)\(0\)\(1\)\(1\)\(3\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(3\)
\(2\)\(0\)\(2\)\(2\)\(4\)\(4\)\(1\)\(2\)\(2\)\(4\)

Note:

This is operator "13.6" from ...

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8

New Number: 13.7 |  AESZ:  |  Superseeker: 10 7709/9  |  Hash: 47093f7f3b7ab4544ef6b418bdae778b  

Degree: 13

\(\theta^4+x\left(127\theta^4-2\theta^3+22\theta^2+23\theta+6\right)+x^{2}\left(4803\theta^4+1644\theta^3+3459\theta^2+430\theta-384\right)+2^{3} x^{3}\left(2507\theta^4+8118\theta^3-2448\theta^2-7127\theta-2940\right)-2^{4} x^{4}\left(94175\theta^4+88358\theta^3+133418\theta^2+111507\theta+38898\right)+2^{6} 3 x^{5}\left(22347\theta^4+197706\theta^3+783766\theta^2+893091\theta+359952\right)+2^{6} 3^{2} x^{6}\left(869067\theta^4+4718208\theta^3+11162457\theta^2+11758320\theta+4583500\right)-2^{9} 3^{3} x^{7}\left(245985\theta^4+1338174\theta^3+3414812\theta^2+4418167\theta+2103502\right)-2^{12} 3^{4} x^{8}\left(234234\theta^4+2167368\theta^3+7012373\theta^2+9416514\theta+4375751\right)+2^{15} 3^{5} x^{9}\left(81234\theta^4+643380\theta^3+1815861\theta^2+2193249\theta+947968\right)+2^{18} 3^{6} x^{10}(\theta+1)(15879\theta^3+214401\theta^2+816191\theta+896789)-2^{21} 3^{7} x^{11}(\theta+1)(\theta+2)(8037\theta^2+71103\theta+151546)+2^{27} 3^{9} x^{12}(\theta+3)(\theta+2)(\theta+1)(31\theta+152)-2^{29} 3^{9} 5 x^{13}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -6, 90, -1368, 21546, ...
--> OEIS
Normalized instanton numbers (n0=1): 10, -149/2, 7709/9, -27333/2, 242829, ... ; Common denominator:...

Discriminant

\(-(16z+1)(2160z^3+27z^2-9z+1)(24z+1)^2(72z^2-48z-1)^2(8z-1)^3\)

Local exponents

≈\(-0.100198\)\(-\frac{ 1}{ 16}\)\(-\frac{ 1}{ 24}\)\(\frac{ 1}{ 3}-\frac{ 1}{ 4}\sqrt{ 2}\)\(0\) ≈\(0.043849-0.05194I\) ≈\(0.043849+0.05194I\)\(\frac{ 1}{ 8}\)\(\frac{ 1}{ 3}+\frac{ 1}{ 4}\sqrt{ 2}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(2\)
\(1\)\(1\)\(\frac{ 1}{ 2}\)\(3\)\(0\)\(1\)\(1\)\(\frac{ 3}{ 2}\)\(3\)\(3\)
\(2\)\(2\)\(1\)\(4\)\(0\)\(2\)\(2\)\(2\)\(4\)\(4\)

Note:

This is operator "13.7" from ...

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9

New Number: 13.8 |  AESZ:  |  Superseeker: 8 -830/9  |  Hash: bcea3fff557004b4da26e9aa34caac6c  

Degree: 13

\(\theta^4-x\left(55\theta^4+142\theta^3+112\theta^2+41\theta+6\right)+x^{2}\left(456\theta^4+4668\theta^3+7455\theta^2+3958\theta+696\right)+x^{3}\left(35078\theta^4+127188\theta^3+175671\theta^2+133507\theta+41718\right)+x^{4}\left(82753\theta^4+664768\theta^3+2450839\theta^2+2316756\theta+736812\right)-3 x^{5}\left(885105\theta^4+1342938\theta^3-883331\theta^2-2706576\theta-1350228\right)-2 3^{2} x^{6}\left(345501\theta^4+3334206\theta^3+4969485\theta^2+2964744\theta+630748\right)+2^{2} 3^{3} x^{7}\left(459939\theta^4+270666\theta^3-1625381\theta^2-2377792\theta-962956\right)+2^{4} 3^{4} x^{8}\left(112581\theta^4+699447\theta^3+1277449\theta^2+1022649\theta+314494\right)-2^{4} 3^{5} x^{9}\left(34101\theta^4-33864\theta^3-473835\theta^2-744726\theta-350272\right)-2^{5} 3^{6} x^{10}(\theta+1)(20847\theta^3+146325\theta^2+303230\theta+217616)+2^{6} 3^{7} x^{11}(\theta+1)(\theta+2)(1791\theta^2-1173\theta-14800)+2^{9} 3^{9} x^{12}(\theta+3)(\theta+2)(\theta+1)(52\theta+257)-2^{10} 3^{9} 17 x^{13}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 90, 1044, -5670, ...
--> OEIS
Normalized instanton numbers (n0=1): 8, -45/2, -830/9, -5301/2, 2790, ... ; Common denominator:...

Discriminant

\(-(2z+1)(3672z^3+1728z^2-72z+1)(6z-1)^2(12z+1)^2(3z+1)^2(z-1)^3\)

Local exponents

≈\(-0.510076\)\(-\frac{ 1}{ 2}\)\(-\frac{ 1}{ 3}\)\(-\frac{ 1}{ 12}\)\(0\) ≈\(0.019744-0.012003I\) ≈\(0.019744+0.012003I\)\(\frac{ 1}{ 6}\)\(1\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(2\)
\(1\)\(1\)\(\frac{ 1}{ 2}\)\(3\)\(0\)\(1\)\(1\)\(3\)\(\frac{ 3}{ 2}\)\(3\)
\(2\)\(2\)\(1\)\(4\)\(0\)\(2\)\(2\)\(4\)\(2\)\(4\)

Note:

This is operator "13.8" from ...

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10

New Number: 13.9 |  AESZ:  |  Superseeker: 8 2200/9  |  Hash: 31ff3b7bd4c8fed070ee43b6903d3752  

Degree: 13

\(\theta^4+2^{3} x\theta(4\theta^3-8\theta^2-5\theta-1)-2^{4} x^{2}\left(48\theta^4+120\theta^3+45\theta^2+74\theta+36\right)-2^{7} x^{3}\left(101\theta^4-342\theta^3-387\theta^2-410\theta-171\right)+2^{8} x^{4}\left(3121\theta^4+14104\theta^3+30889\theta^2+27720\theta+9351\right)+2^{11} 3^{2} x^{5}\left(655\theta^4+4062\theta^3+10081\theta^2+10272\theta+3856\right)-2^{12} 3^{2} x^{6}\left(2272\theta^4+2816\theta^3-9950\theta^2-18768\theta-8813\right)-2^{15} 3^{3} x^{7}\left(1546\theta^4+12172\theta^3+30708\theta^2+33880\theta+13843\right)+2^{16} 3^{4} x^{8}\left(1099\theta^4+1344\theta^3-11134\theta^2-23964\theta-13063\right)+2^{19} 3^{5} x^{9}\left(458\theta^4+4828\theta^3+15325\theta^2+19721\theta+8830\right)-2^{20} 3^{6} x^{10}(\theta+1)(368\theta^3+1752\theta^2+1297\theta-1035)-2^{23} 3^{7} x^{11}(\theta+1)(\theta+2)(39\theta^2+513\theta+1172)+2^{24} 3^{9} x^{12}(\theta+3)(\theta+2)(\theta+1)(17\theta+82)-2^{27} 3^{10} x^{13}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

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Coefficients of the holomorphic solution: 1, 0, 36, -192, -4284, ...
--> OEIS
Normalized instanton numbers (n0=1): 8, -75/2, 2200/9, -8117/2, 47936, ... ; Common denominator:...

Discriminant

\(-(8z+1)(5184z^3+432z^2-36z+1)(12z+1)^2(144z^2-24z-1)^2(4z-1)^3\)

Local exponents

≈\(-0.141868\)\(-\frac{ 1}{ 8}\)\(-\frac{ 1}{ 12}\)\(\frac{ 1}{ 12}-\frac{ 1}{ 12}\sqrt{ 2}\)\(0\) ≈\(0.029267-0.022431I\) ≈\(0.029267+0.022431I\)\(\frac{ 1}{ 12}+\frac{ 1}{ 12}\sqrt{ 2}\)\(\frac{ 1}{ 4}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(2\)
\(1\)\(1\)\(\frac{ 1}{ 2}\)\(3\)\(0\)\(1\)\(1\)\(3\)\(\frac{ 3}{ 2}\)\(3\)
\(2\)\(2\)\(1\)\(4\)\(0\)\(2\)\(2\)\(4\)\(2\)\(4\)

Note:

This is operator "13.9" from ...

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11

New Number: 7.12 |  AESZ:  |  Superseeker: -21 -7941  |  Hash: 0841b278bc566a089b643bbe2460fe8b  

Degree: 7

\(\theta^4+3 x\left(99\theta^4+162\theta^3+139\theta^2+58\theta+10\right)+2 3^{4} x^{2}\left(135\theta^4+738\theta^3+945\theta^2+518\theta+116\right)-2^{2} 3^{7} x^{3}\left(117\theta^4-738\theta^3-2010\theta^2-1493\theta-406\right)-2^{3} 3^{10} x^{4}\left(333\theta^4+774\theta^3-898\theta^2-1269\theta-454\right)-2^{4} 3^{13} x^{5}\left(54\theta^4+1224\theta^3+1179\theta^2+347\theta-22\right)+2^{5} 3^{16} x^{6}\left(180\theta^4+72\theta^3-327\theta^2-359\theta-106\right)+2^{7} 3^{19} x^{7}(\theta+1)^2(6\theta+5)(6\theta+7)\)

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Coefficients of the holomorphic solution: 1, -30, 1458, -89076, 6250050, ...
--> OEIS
Normalized instanton numbers (n0=1): -21, -399, -7941, -986355/4, -8179455, ... ; Common denominator:...

Discriminant

\((27z+1)(54z+1)(54z-1)^2(108z+1)^3\)

Local exponents

\(-\frac{ 1}{ 27}\)\(-\frac{ 1}{ 54}\)\(-\frac{ 1}{ 108}\)\(0\)\(\frac{ 1}{ 54}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 5}{ 6}\)
\(1\)\(1\)\(\frac{ 1}{ 2}\)\(0\)\(1\)\(1\)
\(1\)\(1\)\(\frac{ 3}{ 2}\)\(0\)\(3\)\(1\)
\(2\)\(2\)\(2\)\(0\)\(4\)\(\frac{ 7}{ 6}\)

Note:

This is operator "7.12" from ...

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12

New Number: 7.15 |  AESZ:  |  Superseeker: -2 -952  |  Hash: d9a911258d890c112974a4ba19e93e6d  

Degree: 7

\(\theta^4+2 x\left(138\theta^4+156\theta^3+137\theta^2+59\theta+10\right)+2^{2} x^{2}\left(4796\theta^4+15824\theta^3+16719\theta^2+7610\theta+1400\right)-2^{4} 5 x^{3}\left(5876\theta^4-28824\theta^3-58439\theta^2-39075\theta-9350\right)-2^{6} 5 x^{4}\left(184592\theta^4+414976\theta^3-60816\theta^2-180968\theta-60145\right)+2^{10} 3 x^{5}\left(240624\theta^4-905760\theta^3-1250920\theta^2-576920\theta-83925\right)+2^{18} 3^{2} x^{6}\left(13608\theta^4+48276\theta^3+66402\theta^2+41679\theta+9935\right)-2^{20} 3^{5} x^{7}(6\theta+5)^2(6\theta+7)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -20, 900, -55280, 3962500, ...
--> OEIS
Normalized instanton numbers (n0=1): -2, -343/2, -952, -45148, -17303644/25, ... ; Common denominator:...

Discriminant

\(-(-1-16z+256z^2)(32z-1)^2(108z+1)^3\)

Local exponents

\(\frac{ 1}{ 32}-\frac{ 1}{ 32}\sqrt{ 5}\)\(-\frac{ 1}{ 108}\)\(0\)\(\frac{ 1}{ 32}\)\(\frac{ 1}{ 32}+\frac{ 1}{ 32}\sqrt{ 5}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 5}{ 6}\)
\(1\)\(\frac{ 1}{ 2}\)\(0\)\(1\)\(1\)\(\frac{ 5}{ 6}\)
\(1\)\(\frac{ 3}{ 2}\)\(0\)\(3\)\(1\)\(\frac{ 7}{ 6}\)
\(2\)\(2\)\(0\)\(4\)\(2\)\(\frac{ 7}{ 6}\)

Note:

This is operator "7.15" from ...

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13

New Number: 7.1 |  AESZ:  |  Superseeker: 10/7 508/7  |  Hash: 08ab3cb496250adfa30bc3e24ac63c4f  

Degree: 7

\(7^{2} \theta^4-2 7 x\theta(46\theta^3+52\theta^2+33\theta+7)-2^{2} x^{2}\left(7332\theta^4+28848\theta^3+42633\theta^2+26670\theta+6272\right)-2^{4} x^{3}\left(2860\theta^4+44760\theta^3+120483\theta^2+111279\theta+35098\right)+2^{9} x^{4}\left(2230\theta^4+5920\theta^3-741\theta^2-6509\theta-3049\right)+2^{14} x^{5}\left(174\theta^4+1320\theta^3+1971\theta^2+1095\theta+190\right)-2^{19} x^{6}\left(22\theta^4+24\theta^3-9\theta^2-21\theta-7\right)-2^{25} x^{7}\left((\theta+1)^4\right)\)

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Coefficients of the holomorphic solution: 1, 0, 32, 288, 7776, ...
--> OEIS
Normalized instanton numbers (n0=1): 10/7, 100/7, 508/7, 808, 59910/7, ... ; Common denominator:...

Discriminant

\(-(16z+1)(32z-1)(32z-7)^2(4z+1)^3\)

Local exponents

\(-\frac{ 1}{ 4}\)\(-\frac{ 1}{ 16}\)\(0\)\(\frac{ 1}{ 32}\)\(\frac{ 7}{ 32}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(\frac{ 1}{ 2}\)\(1\)\(0\)\(1\)\(1\)\(1\)
\(\frac{ 3}{ 2}\)\(1\)\(0\)\(1\)\(3\)\(1\)
\(2\)\(2\)\(0\)\(2\)\(4\)\(1\)

Note:

This is operator "7.1" from ...

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14

New Number: 7.2 |  AESZ:  |  Superseeker: -80 -249872  |  Hash: 341389ebf4ab0242c5b70d9a8fd7a1d9  

Degree: 7

\(\theta^4+2^{4} x\left(22\theta^4+64\theta^3+51\theta^2+19\theta+3\right)-2^{9} x^{2}\left(174\theta^4-624\theta^3-945\theta^2-417\theta-80\right)-2^{14} x^{3}\left(2230\theta^4+3000\theta^3-5121\theta^2-3813\theta-971\right)+2^{19} x^{4}\left(2860\theta^4-33320\theta^3+3363\theta^2+6847\theta+2402\right)+2^{27} x^{5}\left(7332\theta^4+480\theta^3+81\theta^2+1380\theta+719\right)+2^{36} 7 x^{6}(\theta+1)(46\theta^3+86\theta^2+67\theta+20)-2^{45} 7^{2} x^{7}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -48, 5072, -733440, 124117776, ...
--> OEIS
Normalized instanton numbers (n0=1): -80, -4202, -249872, -22251117, -2195810928, ... ; Common denominator:...

Discriminant

\(-(64z+1)(32z-1)(224z-1)^2(256z+1)^3\)

Local exponents

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

Note:

This is operator "7.2" from ...

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15

New Number: 8.79 |  AESZ:  |  Superseeker: 22/5 68  |  Hash: 064e5b590dd8b6a4daa1e905fbe693c2  

Degree: 8

\(5^{2} \theta^4-2 5 x\left(338\theta^4+412\theta^3+371\theta^2+165\theta+30\right)+2^{2} x^{2}\left(46396\theta^4+103408\theta^3+125291\theta^2+76370\theta+19080\right)-2^{4} 3 x^{3}\left(115508\theta^4+357896\theta^3+524149\theta^2+375205\theta+106530\right)+2^{6} 3^{2} x^{4}\left(173456\theta^4+669024\theta^3+1118292\theta^2+883484\theta+269049\right)-2^{11} 3^{3} x^{5}\left(20272\theta^4+91616\theta^3+168594\theta^2+142006\theta+45053\right)+2^{14} 3^{4} x^{6}\left(5792\theta^4+29504\theta^3+58300\theta^2+51220\theta+16641\right)-2^{21} 3^{5} x^{7}(\theta+1)^2(58\theta^2+208\theta+201)+2^{26} 3^{6} x^{8}(\theta+1)^2(\theta+2)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 204, 4368, 112140, ...
--> OEIS
Normalized instanton numbers (n0=1): 22/5, 8, 68, 3292/5, 38826/5, ... ; Common denominator:...

Discriminant

\((-1+48z)(16z-1)^2(48z-5)^2(12z-1)^3\)

Local exponents

\(0\)\(\frac{ 1}{ 48}\)\(\frac{ 1}{ 16}\)\(\frac{ 1}{ 12}\)\(\frac{ 5}{ 48}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(\frac{ 1}{ 2}\)\(1\)\(1\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(\frac{ 3}{ 2}\)\(3\)\(2\)
\(0\)\(2\)\(1\)\(2\)\(4\)\(2\)

Note:

This is operator "8.79" from ...

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16

New Number: 9.3 |  AESZ:  |  Superseeker: 10 3394/3  |  Hash: 40e3715abcc5c4cb07e700ca79f80abf  

Degree: 9

\(\theta^4-x\left(57\theta^4+116\theta^3+84\theta^2+26\theta+3\right)-2 x^{2}\left(894\theta^4+3208\theta^3+4571\theta^2+2771\theta+651\right)-2 x^{3}\left(7322\theta^4+56368\theta^3+124783\theta^2+101099\theta+29757\right)+2 3^{2} x^{4}\left(6967\theta^4-27080\theta^3-139991\theta^2-138507\theta-45297\right)+2 3^{4} x^{5}\left(17617\theta^4+49068\theta^3-31255\theta^2-79893\theta-34578\right)+2 3^{8} x^{6}\left(1082\theta^4+8360\theta^3+7967\theta^2+1439\theta-773\right)-2 3^{11} x^{7}\left(198\theta^4-864\theta^3-1545\theta^2-909\theta-155\right)-3^{15} x^{8}\left(69\theta^4+144\theta^3+126\theta^2+54\theta+10\right)-3^{20} x^{9}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 3, 135, 5349, 258039, ...
--> OEIS
Normalized instanton numbers (n0=1): 10, 77, 3394/3, 24029, 640402, ... ; Common denominator:...

Discriminant

\(-(-1+81z)(-1+9z)^2(81z^2+14z+1)^3\)

Local exponents

\(-\frac{ 7}{ 81}-\frac{ 4}{ 81}\sqrt{ 2}I\)\(-\frac{ 7}{ 81}+\frac{ 4}{ 81}\sqrt{ 2}I\)\(0\)\(\frac{ 1}{ 81}\)\(\frac{ 1}{ 9}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(\frac{ 1}{ 2}\)\(\frac{ 1}{ 2}\)\(0\)\(1\)\(1\)\(1\)
\(\frac{ 3}{ 2}\)\(\frac{ 3}{ 2}\)\(0\)\(1\)\(3\)\(1\)
\(2\)\(2\)\(0\)\(2\)\(4\)\(1\)

Note:

This is operator "9.3" from ...

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17

New Number: 9.4 |  AESZ:  |  Superseeker: -90 -413926  |  Hash: e2329b2f9cd1e3f65d29644e6ce39d24  

Degree: 9

\(\theta^4+3^{2} x\left(69\theta^4+132\theta^3+108\theta^2+42\theta+7\right)+2 3^{5} x^{2}\left(198\theta^4+1656\theta^3+2235\theta^2+1203\theta+271\right)-2 3^{9} x^{3}\left(1082\theta^4-4032\theta^3-10621\theta^2-6257\theta-1523\right)-2 3^{12} x^{4}\left(17617\theta^4+21400\theta^3-72757\theta^2-59353\theta-17391\right)-2 3^{17} x^{5}\left(6967\theta^4+54948\theta^3-16949\theta^2-32367\theta-12734\right)+2 3^{22} x^{6}\left(7322\theta^4-27080\theta^3-389\theta^2+8651\theta+4395\right)+2 3^{29} x^{7}\left(894\theta^4+368\theta^3+311\theta^2+323\theta+137\right)+3^{36} x^{8}\left(57\theta^4+112\theta^3+78\theta^2+22\theta+2\right)-3^{43} x^{9}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -63, 4455, 34551, -114913161, ...
--> OEIS
Normalized instanton numbers (n0=1): -90, -8685/2, -413926, -38862153, -4502063682, ... ; Common denominator:...

Discriminant

\(-(-1+27z)(-1+243z)^2(59049z^2+378z+1)^3\)

Local exponents

\(-\frac{ 7}{ 2187}-\frac{ 4}{ 2187}\sqrt{ 2}I\)\(-\frac{ 7}{ 2187}+\frac{ 4}{ 2187}\sqrt{ 2}I\)\(0\)\(\frac{ 1}{ 243}\)\(\frac{ 1}{ 27}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(1\)
\(\frac{ 1}{ 2}\)\(\frac{ 1}{ 2}\)\(0\)\(1\)\(1\)\(1\)
\(\frac{ 3}{ 2}\)\(\frac{ 3}{ 2}\)\(0\)\(3\)\(1\)\(1\)
\(2\)\(2\)\(0\)\(4\)\(2\)\(1\)

Note:

This is operator "9.4" from ...

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