### Summary

You searched for: sol=-4

1

New Number: 3.8 |  AESZ: ~100  |  Superseeker: 5 454  |  Hash: 82a1ac6ac6fb9ab2e4d6b5d5790d1d9b

Degree: 3

$\theta^4+x\left(15\theta^4+30\theta^3+35\theta^2+20\theta+4\right)-2^{5} x^{2}(\theta+1)^2(66\theta^2+132\theta+53)-2^{8} 7^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)$

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Coefficients of the holomorphic solution: 1, -4, 132, -1120, 72100, ...
--> OEIS
Normalized instanton numbers (n0=1): 5, 42, 454, 7498, 154351, ... ; Common denominator:...

#### Discriminant

$-(49z-1)(1+32z)^2$

#### Local exponents

$-\frac{ 1}{ 32}$$0$$\frac{ 1}{ 49}$$\infty$
$0$$0$$0$$\frac{ 1}{ 2}$
$\frac{ 1}{ 2}$$0$$1$$1$
$\frac{ 1}{ 2}$$0$$1$$2$
$1$$0$$2$$\frac{ 5}{ 2}$

#### Note:

Operator equivalent to AESZ 100= $a \ast a$

2

New Number: 5.31 |  AESZ: 212  |  Superseeker: -20/7 -104  |  Hash: f72aa947ba945355102b3fef56e0af0f

Degree: 5

$7^{2} \theta^4+2 7 x\left(134\theta^4+286\theta^3+234\theta^2+91\theta+14\right)-2^{2} x^{2}\left(3183\theta^4+10266\theta^3+13501\theta^2+8225\theta+1918\right)+2^{3} x^{3}\left(2588\theta^4+8400\theta^3+10256\theta^2+5649\theta+1190\right)-2^{4} 3 x^{4}\left(256\theta^4+848\theta^3+1141\theta^2+717\theta+174\right)+2^{8} 3^{2} x^{5}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -4, 64, -1408, 37216, ...
--> OEIS
Normalized instanton numbers (n0=1): -20/7, 57/4, -104, 16385/14, -110508/7, ... ; Common denominator:...

#### Discriminant

$(4z-1)(16z^2-44z-1)(6z-7)^2$

#### Local exponents

$\frac{ 11}{ 8}-\frac{ 5}{ 8}\sqrt{ 5}$$0$$\frac{ 1}{ 4}$$\frac{ 7}{ 6}$$\frac{ 11}{ 8}+\frac{ 5}{ 8}\sqrt{ 5}$$\infty$
$0$$0$$0$$0$$0$$1$
$1$$0$$1$$1$$1$$1$
$1$$0$$1$$3$$1$$1$
$2$$0$$2$$4$$2$$1$

#### Note:

There is a second MUM-point corresponding to Operator AESZ 117 /5.515.

3

New Number: 10.6 |  AESZ:  |  Superseeker: 8 2200/9  |  Hash: b5aa0abf76ddfbd280ec220a43822aa4

Degree: 10

$\theta^4+2^{2} x\left(21\theta^4-6\theta^3+3\theta+1\right)+2^{4} x^{2}\left(126\theta^4-96\theta^3-16\theta^2-56\theta-33\right)+2^{6} x^{3}\left(84\theta^4-336\theta^3-226\theta^2-366\theta-163\right)+2^{11} 3 x^{4}\left(39\theta^4+500\theta^3+1230\theta^2+1160\theta+407\right)+2^{12} x^{5}\left(7029\theta^4+50118\theta^3+125086\theta^2+129149\theta+48902\right)+2^{14} x^{6}\left(38550\theta^4+294456\theta^3+806428\theta^2+911232\theta+368273\right)+2^{16} x^{7}\left(77544\theta^4+708720\theta^3+2233434\theta^2+2804346\theta+1214177\right)+2^{20} x^{8}\left(9171\theta^4+117228\theta^3+467444\theta^2+684316\theta+324572\right)-2^{23} x^{9}(2\theta+3)(2114\theta^3+16713\theta^2+37111\theta+22497)+2^{26} 3 5^{2} x^{10}(\theta+1)(2\theta+5)(2\theta+3)(\theta+3)$

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Coefficients of the holomorphic solution: 1, -4, 52, -688, 2500, ...
--> OEIS
Normalized instanton numbers (n0=1): 8, -75/2, 2200/9, -8117/2, 47936, ... ; Common denominator:...

#### Discriminant

$(12z+1)(6400z^3+192z^2-24z+1)(16z+1)^2(32z^2-32z-1)^2$

#### Local exponents

≈$-0.090507$$-\frac{ 1}{ 12}$$-\frac{ 1}{ 16}$$\frac{ 1}{ 2}-\frac{ 3}{ 8}\sqrt{ 2}$$0$ ≈$0.030254-0.02848I$ ≈$0.030254+0.02848I$$\frac{ 1}{ 2}+\frac{ 3}{ 8}\sqrt{ 2}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$0$$1$
$1$$1$$\frac{ 1}{ 2}$$1$$0$$1$$1$$1$$\frac{ 3}{ 2}$
$1$$1$$\frac{ 1}{ 2}$$3$$0$$1$$1$$3$$\frac{ 5}{ 2}$
$2$$2$$1$$4$$0$$2$$2$$4$$3$

#### Note:

This is operator "10.6" from ...

4

New Number: 12.3 |  AESZ:  |  Superseeker: -12/5 444/5  |  Hash: 45726409a4c817f929c9e6e49b33a941

Degree: 12

$5^{2} \theta^4+2^{2} 5 x\left(4\theta^4+56\theta^3+53\theta^2+25\theta+5\right)-2^{4} x^{2}\left(976\theta^4+6208\theta^3+9016\theta^2+6360\theta+1985\right)+2^{8} x^{3}\left(832\theta^4-2304\theta^3-11276\theta^2-12780\theta-5495\right)+2^{13} x^{4}\left(176\theta^4+4672\theta^3+16244\theta^2+19860\theta+9145\right)-2^{16} x^{5}\left(1824\theta^4+8448\theta^3+1052\theta^2-6884\theta-5771\right)+2^{21} x^{6}\left(432\theta^4+192\theta^3-3816\theta^2-9540\theta-5869\right)+2^{24} x^{7}\left(704\theta^4+10048\theta^3+21804\theta^2+22348\theta+7847\right)-2^{29} x^{8}\left(472\theta^4+2176\theta^3+7884\theta^2+11644\theta+5965\right)+2^{32} x^{9}\left(336\theta^4+672\theta^3+1144\theta^2+2904\theta+2145\right)+2^{36} x^{10}\left(368\theta^4+1216\theta^3+1304\theta^2-240\theta-697\right)-2^{44} x^{11}(2\theta+3)(4\theta^3+28\theta^2+51\theta+28)-2^{46} x^{12}\left((2\theta+3)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -4, 108, -912, 21484, ...
--> OEIS
Normalized instanton numbers (n0=1): -12/5, 103/5, 444/5, 1148/5, -6704, ... ; Common denominator:...

#### Discriminant

$-(-1-16z+256z^2)(16z+1)^2(16z-1)^2(8192z^3+768z^2-32z+5)^2$

#### Local exponents

≈$-0.148005$$-\frac{ 1}{ 16}$$\frac{ 1}{ 32}-\frac{ 1}{ 32}\sqrt{ 5}$$0$ ≈$0.027128-0.058206I$ ≈$0.027128+0.058206I$$\frac{ 1}{ 16}$$\frac{ 1}{ 32}+\frac{ 1}{ 32}\sqrt{ 5}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$0$$\frac{ 3}{ 2}$
$1$$0$$1$$0$$1$$1$$\frac{ 1}{ 2}$$1$$\frac{ 3}{ 2}$
$3$$1$$1$$0$$3$$3$$\frac{ 1}{ 2}$$1$$\frac{ 3}{ 2}$
$4$$1$$2$$0$$4$$4$$1$$2$$\frac{ 3}{ 2}$

#### Note:

This is operator "12.3" from ...

5

New Number: 7.19 |  AESZ:  |  Superseeker: 4/3 -124/81  |  Hash: f7f0f5d883101c38ed22cb74c80c8f5c

Degree: 7

$3^{3} \theta^4-2^{2} 3^{2} x\left(12\theta^4-16\theta^3-21\theta^2-13\theta-3\right)-2^{4} 3 x^{2}\left(240\theta^4+1280\theta^3+2068\theta^2+1636\theta+489\right)+2^{10} x^{3}\left(212\theta^4+592\theta^3+490\theta^2-34\theta-129\right)+2^{13} x^{4}\left(72\theta^4+1536\theta^3+4804\theta^2+5468\theta+2031\right)-2^{18} x^{5}\left(100\theta^4+720\theta^3+1535\theta^2+1155\theta+276\right)+2^{20} x^{6}\left(144\theta^4+768\theta^3+1268\theta^2+884\theta+229\right)-2^{28} x^{7}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -4, 68, -496, 9796, ...
--> OEIS
Normalized instanton numbers (n0=1): 4/3, -14/9, -124/81, -4498/243, 37024/729, ... ; Common denominator:...

#### Discriminant

$-(16z-1)^2(16z+1)^2(16z-3)^3$

#### Local exponents

$-\frac{ 1}{ 16}$$0$$\frac{ 1}{ 16}$$\frac{ 3}{ 16}$$\infty$
$0$$0$$0$$0$$1$
$0$$0$$\frac{ 1}{ 2}$$2$$1$
$1$$0$$\frac{ 1}{ 2}$$3$$1$
$1$$0$$1$$5$$1$

#### Note:

This is operator "7.19" from ...

6

New Number: 7.20 |  AESZ:  |  Superseeker: 10 3394/3  |  Hash: 9d5791eaabb9d0e9cb4b5cd0b2158b12

Degree: 7

$\theta^4-x\left(88\theta^3-4+71\theta^4+42\theta^2-2\theta\right)-x^{2}\left(10462\theta+13294\theta^2+875\theta^4+6848\theta^3+3132\right)+3^{2} x^{3}\left(373\theta^4-6360\theta^3-30716\theta^2-44868\theta-23180\right)+3^{4} x^{4}\left(1843\theta^4+8384\theta^3+3236\theta^2-14996\theta-15180\right)+3^{8} x^{5}\left(75\theta^4+1272\theta^3+3454\theta^2+3554\theta+1192\right)-3^{11} x^{6}\left(27\theta^4-414\theta^2-918\theta-584\right)-3^{16} x^{7}\left((\theta+2)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -4, 147, 4496, 223111, ...
--> OEIS
Normalized instanton numbers (n0=1): 10, 77, 3394/3, 24029, 640402, ... ; Common denominator:...

#### Discriminant

$-(-1+81z)(9z-1)^2(81z^2+14z+1)^2$

#### 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$$0$$2$
$-\frac{ 1}{ 2}$$-\frac{ 1}{ 2}$$0$$1$$1$$2$
$1$$1$$0$$1$$3$$2$
$\frac{ 3}{ 2}$$\frac{ 3}{ 2}$$0$$2$$4$$2$

#### Note:

This is operator "7.20" from ...

7

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

8

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 ...

9

New Number: 8.65 |  AESZ:  |  Superseeker: -24/5 -1608/5  |  Hash: 5e457fa5807a784e24220c973aeceba8

Degree: 8

$5^{2} \theta^4+2^{2} 5 x\left(73\theta^4+122\theta^3+96\theta^2+35\theta+5\right)-2^{4} x^{2}\left(134\theta^4+2072\theta^3+3924\theta^2+2660\theta+645\right)-2^{6} x^{3}\left(708\theta^4+672\theta^3-2898\theta^2-3750\theta-1285\right)+2^{10} x^{4}\left(110\theta^4+700\theta^3+498\theta^2-56\theta-105\right)+2^{12} x^{5}\left(61\theta^4-266\theta^3-544\theta^2-373\theta-88\right)-2^{14} x^{6}\left(86\theta^4+168\theta^3+172\theta^2+108\theta+31\right)+2^{16} x^{7}\left(32\theta^4+112\theta^3+158\theta^2+102\theta+25\right)-2^{20} x^{8}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -4, 92, -2704, 95596, ...
--> OEIS
Normalized instanton numbers (n0=1): -24/5, 329/10, -1608/5, 48409/10, -455264/5, ... ; Common denominator:...

#### Discriminant

$-(4z-1)(256z^3-192z^2+56z+1)(-5-16z+32z^2)^2$

#### Local exponents

$\frac{ 1}{ 4}-\frac{ 1}{ 8}\sqrt{ 14}$ ≈$-0.016861$$0$$\frac{ 1}{ 4}$ ≈$0.38343-0.290965I$ ≈$0.38343+0.290965I$$\frac{ 1}{ 4}+\frac{ 1}{ 8}\sqrt{ 14}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$1$
$1$$1$$0$$1$$1$$1$$1$$1$
$3$$1$$0$$1$$1$$1$$3$$1$
$4$$2$$0$$2$$2$$2$$4$$1$

#### Note:

This is operator "8.65" from ...