### Summary

You searched for: sol=6

1

New Number: 2.54 |  AESZ: 41  |  Superseeker: 2 -104  |  Hash: a9ddeed4299f59fb9ac9f6f248383b8f

Degree: 2

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

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 54, 60, -19530, ...
--> OEIS
Normalized instanton numbers (n0=1): 2, -7, -104, -588, 3300, ... ; Common denominator:...

#### Discriminant

$1-56z+1296z^2$

#### Local exponents

$0$$\frac{ 7}{ 324}-\frac{ 1}{ 81}\sqrt{ 2}I$$\frac{ 7}{ 324}+\frac{ 1}{ 81}\sqrt{ 2}I$$\infty$
$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$1$
$0$$1$$1$$1$
$0$$2$$2$$\frac{ 3}{ 2}$

#### Note:

Hadamard product $I \ast \delta$

2

New Number: 2.56 |  AESZ: 185  |  Superseeker: 6 608  |  Hash: 80506439e4d4fdc41f5b16e246a69fbf

Degree: 2

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

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 162, 6180, 284130, ...
--> OEIS
Normalized instanton numbers (n0=1): 6, 93/2, 608, 11754, 275352, ... ; Common denominator:...

#### Discriminant

$1-72z-432z^2$

#### Local exponents

$-\frac{ 1}{ 12}-\frac{ 1}{ 18}\sqrt{ 3}$$0$$-\frac{ 1}{ 12}+\frac{ 1}{ 18}\sqrt{ 3}$$\infty$
$0$$0$$0$$\frac{ 1}{ 2}$
$1$$0$$1$$1$
$1$$0$$1$$1$
$2$$0$$2$$\frac{ 3}{ 2}$

#### Note:

Hadamard product $I \ast \zeta$

3

New Number: 2.62 |  AESZ: 28  |  Superseeker: 5 312  |  Hash: 06dd455cafc5097e4f671d385984c1a2

Degree: 2

$\theta^4-x\left(65\theta^4+130\theta^3+105\theta^2+40\theta+6\right)+2^{2} x^{2}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 126, 3948, 149310, ...
--> OEIS
Normalized instanton numbers (n0=1): 5, 28, 312, 4808, 91048, ... ; Common denominator:...

#### Discriminant

$(64z-1)(z-1)$

#### Local exponents

$0$$\frac{ 1}{ 64}$$1$$\infty$
$0$$0$$0$$\frac{ 3}{ 4}$
$0$$1$$1$$1$
$0$$1$$1$$1$
$0$$2$$2$$\frac{ 5}{ 4}$

#### Note:

A-incarnation: $X(1, 1, 1, 1, 1, 1) \subset Grass(3, 6)$

4

New Number: 2.64 |  AESZ: 182  |  Superseeker: 1 7  |  Hash: 89ba4729efa82413b33fe6928ff8d2c9

Degree: 2

$\theta^4-x\left(43\theta^4+86\theta^3+77\theta^2+34\theta+6\right)+2^{2} 3 x^{2}(\theta+1)^2(6\theta+5)(6\theta+7)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 66, 924, 14850, ...
--> OEIS
Normalized instanton numbers (n0=1): 1, 7/4, 7, 40, 270, ... ; Common denominator:...

#### Discriminant

$(27z-1)(16z-1)$

#### Local exponents

$0$$\frac{ 1}{ 27}$$\frac{ 1}{ 16}$$\infty$
$0$$0$$0$$\frac{ 5}{ 6}$
$0$$1$$1$$1$
$0$$1$$1$$1$
$0$$2$$2$$\frac{ 7}{ 6}$

#### Note:

This is operator "2.64" from ...

5

New Number: 5.102 |  AESZ: 352  |  Superseeker: 1 -12  |  Hash: fc8b141522720827b1dd2cd28a232c1b

Degree: 5

$\theta^4-x\left(70\theta^4+86\theta^3+77\theta^2+34\theta+6\right)+3 x^{2}\left(675\theta^4+1602\theta^3+1933\theta^2+1130\theta+258\right)-2^{2} 3^{3} x^{3}\left(271\theta^4+888\theta^3+1259\theta^2+831\theta+207\right)+2^{2} 3^{5} x^{4}\left(212\theta^4+808\theta^3+1189\theta^2+773\theta+186\right)-2^{4} 3^{7} x^{5}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 54, 492, 3510, ...
--> OEIS
Normalized instanton numbers (n0=1): 1, -7/8, -12, -131/4, 90, ... ; Common denominator:...

#### Discriminant

$-(16z-1)(432z^2-36z+1)(-1+9z)^2$

#### Local exponents

$0$$\frac{ 1}{ 24}-\frac{ 1}{ 72}\sqrt{ 3}I$$\frac{ 1}{ 24}+\frac{ 1}{ 72}\sqrt{ 3}I$$\frac{ 1}{ 16}$$\frac{ 1}{ 9}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 3}{ 4}$
$0$$1$$1$$1$$1$$1$
$0$$1$$1$$1$$3$$1$
$0$$2$$2$$2$$4$$\frac{ 5}{ 4}$

#### Note:

This is operator "5.102" from ...

6

New Number: 5.111 |  AESZ: 380  |  Superseeker: 12 2320  |  Hash: 85214e3836a67470a05358a4d38fb124

Degree: 5

$\theta^4-2 x\left(60\theta^4+90\theta^3+68\theta^2+23\theta+3\right)+2^{2} x^{2}\left(313\theta^4-398\theta^3-1417\theta^2-1033\theta-252\right)+2^{3} x^{3}\left(654\theta^4+5064\theta^3+3574\theta^2+129\theta-405\right)-2^{4} 5 x^{4}\left(628\theta^4-40\theta^3-1699\theta^2-1661\theta-480\right)-2^{6} 3 5^{2} x^{5}(\theta+1)^2(6\theta+5)(6\theta+7)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 246, 13020, 832950, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, 511/4, 2320, 63507, 2180312, ... ; Common denominator:...

#### Discriminant

$-(108z-1)(4z+1)^2(10z-1)^2$

#### Local exponents

$-\frac{ 1}{ 4}$$0$$\frac{ 1}{ 108}$$\frac{ 1}{ 10}$$\infty$
$0$$0$$0$$0$$\frac{ 5}{ 6}$
$\frac{ 1}{ 2}$$0$$1$$1$$1$
$\frac{ 1}{ 2}$$0$$1$$3$$1$
$1$$0$$2$$4$$\frac{ 7}{ 6}$

#### Note:

This is operator "5.111" from ...

7

New Number: 5.122 |  AESZ:  |  Superseeker: 19 18641/3  |  Hash: 7cc1a0411f21ffd93f1a9f6468627432

Degree: 5

$\theta^4+x\left(119\theta^4-194\theta^3-143\theta^2-46\theta-6\right)-2^{2} 3^{2} x^{2}\left(46\theta^4+748\theta^3+379\theta^2+150\theta+27\right)-2^{2} 3^{4} x^{3}\left(2164\theta^4+6264\theta^3+7421\theta^2+4131\theta+846\right)-2^{5} 3^{8} x^{4}(2\theta+1)(76\theta^3+222\theta^2+235\theta+85)-2^{8} 3^{12} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 162, 7620, 334530, ...
--> OEIS
Normalized instanton numbers (n0=1): 19, -170, 18641/3, -163734, 6446745, ... ; Common denominator:...

#### Discriminant

$-(81z-1)(1296z^2+56z+1)(1+72z)^2$

#### Local exponents

$-\frac{ 7}{ 324}-\frac{ 1}{ 81}\sqrt{ 2}I$$-\frac{ 7}{ 324}+\frac{ 1}{ 81}\sqrt{ 2}I$$-\frac{ 1}{ 72}$$0$$\frac{ 1}{ 81}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$1$$1$$0$$1$$1$
$1$$1$$3$$0$$1$$1$
$2$$2$$4$$0$$2$$\frac{ 3}{ 2}$

#### Note:

B-Incarnation as fibre product 62211- 623--1

8

New Number: 5.77 |  AESZ: 307  |  Superseeker: 69/11 8883/11  |  Hash: 3a2dcd4c59d8fa5b7c57250efeecba62

Degree: 5

$11^{2} \theta^4-3 11 x\left(361\theta^4+530\theta^3+419\theta^2+154\theta+22\right)+2^{2} x^{2}\left(47008\theta^4+45904\theta^3-3251\theta^2-17094\theta-4851\right)-2^{4} 3 x^{3}\left(31436\theta^4+86856\theta^3+160363\theta^2+122133\theta+30294\right)+2^{9} 3^{2} x^{4}(2\theta+1)(1252\theta^3+5442\theta^2+6767\theta+2625)-2^{14} 3^{6} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 162, 6540, 314370, ...
--> OEIS
Normalized instanton numbers (n0=1): 69/11, 620/11, 8883/11, 171916/11, 4334406/11, ... ; Common denominator:...

#### Discriminant

$-(81z-1)(64z^2+1)(-11+96z)^2$

#### Local exponents

$0-\frac{ 1}{ 8}I$$0$$0+\frac{ 1}{ 8}I$$\frac{ 1}{ 81}$$\frac{ 11}{ 96}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$0$$1$$1$$1$$1$
$1$$0$$1$$1$$3$$1$
$2$$0$$2$$2$$4$$\frac{ 3}{ 2}$

#### Note:

This is operator "5.77" from ...

9

New Number: 5.78 |  AESZ: 308  |  Superseeker: 248/29 38708/29  |  Hash: 94e96c5d238b2d22a633f4e05ec1ae9f

Degree: 5

$29^{2} \theta^4-2 29 x\left(1318\theta^4+2336\theta^3+1806\theta^2+638\theta+87\right)-2^{2} x^{2}\left(90996\theta^4+744384\theta^3+1267526\theta^2+791584\theta+168345\right)+2^{2} 5^{2} x^{3}\left(34172\theta^4+77256\theta^3-46701\theta^2-110403\theta-36540\right)+2^{4} 5^{4} x^{4}(2\theta+1)(68\theta^3+1842\theta^2+2899\theta+1215)-2^{6} 5^{7} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 210, 9780, 551250, ...
--> OEIS
Normalized instanton numbers (n0=1): 248/29, 2476/29, 38708/29, 940480/29, 27926248/29, ... ; Common denominator:...

#### Discriminant

$-(2000z^3+1024z^2+84z-1)(-29+100z)^2$

#### Local exponents

≈$-0.40534$ ≈$-0.117186$$0$ ≈$0.010526$$\frac{ 29}{ 100}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$1$$0$$1$$1$$1$
$1$$1$$0$$1$$3$$1$
$2$$2$$0$$2$$4$$\frac{ 3}{ 2}$

#### Note:

This is operator "5.78" from ...

10

New Number: 5.94 |  AESZ: 334  |  Superseeker: 7/3 -4843/81  |  Hash: 1ab1dce2847b14dd89a8f8f48ddc7214

Degree: 5

$3^{2} \theta^4-3 x\left(166\theta^4+320\theta^3+271\theta^2+111\theta+18\right)+x^{2}\left(11155\theta^4+42652\theta^3+60463\theta^2+36876\theta+8172\right)-3^{2} x^{3}\left(4705\theta^4+23418\theta^3+42217\theta^2+31152\theta+7932\right)+2^{2} 3 x^{4}\left(3514\theta^4+16403\theta^3+25581\theta^2+16442\theta+3744\right)-2^{2} 5 x^{5}(5\theta+3)(5\theta+4)(5\theta+6)(5\theta+7)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 54, 240, -9450, ...
--> OEIS
Normalized instanton numbers (n0=1): 7/3, -79/12, -4843/81, -1058/3, 3620/3, ... ; Common denominator:...

#### Discriminant

$-(3125z^3-1167z^2+54z-1)(2z-3)^2$

#### Local exponents

$0$ ≈$0.025215-0.018839I$ ≈$0.025215+0.018839I$ ≈$0.32301$$\frac{ 3}{ 2}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 3}{ 5}$
$0$$1$$1$$1$$1$$\frac{ 4}{ 5}$
$0$$1$$1$$1$$3$$\frac{ 6}{ 5}$
$0$$2$$2$$2$$4$$\frac{ 7}{ 5}$

#### Note:

This is operator "5.94" from ...

11

New Number: 11.13 |  AESZ:  |  Superseeker: 70/13 15323/39  |  Hash: 89df09ff1ec0d5dfcae0791579c9095e

Degree: 11

$13^{2} \theta^4-2 13 x\left(593\theta^4+850\theta^3+685\theta^2+260\theta+39\right)+2^{2} x^{2}\left(81227\theta^4+145178\theta^3+121774\theta^2+52312\theta+9477\right)-x^{3}\left(3180153\theta^4+8754414\theta^3+11733109\theta^2+7260552\theta+1687608\right)+2 x^{4}\left(9121117\theta^4+38823752\theta^3+61935546\theta^2+41745416\theta+10192764\right)-2^{2} x^{5}\left(14736265\theta^4+81359956\theta^3+152008790\theta^2+112521671\theta+29176827\right)+2^{2} 3^{2} x^{6}\left(1220244\theta^4+12211662\theta^3+31283769\theta^2+26817500\theta+7548762\right)+2^{2} 3^{2} x^{7}\left(4505067\theta^4+14797690\theta^3+6324743\theta^2-4986206\theta-2940402\right)-2^{3} 3^{3} x^{8}\left(855097\theta^4+3900198\theta^3+2679311\theta^2-619598\theta-662876\right)-2^{4} 3^{3} x^{9}\left(254021\theta^4+398518\theta^3+352691\theta^2+205022\theta+53940\right)+2^{5} 3^{3} 11 x^{10}\left(13283\theta^4+25990\theta^3+18039\theta^2+5062\theta+456\right)+2^{7} 3^{3} 11^{2} x^{11}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 126, 4092, 160110, ...
--> OEIS
Normalized instanton numbers (n0=1): 70/13, 420/13, 15323/39, 78225/13, 1564284/13, ... ; Common denominator:...

#### Discriminant

$(192z^2-69z+1)(2z^3+39z^2-5z+1)(13-112z-18z^2+132z^3)^2$

#### Local exponents

≈$-19.628663$ ≈$-0.912176$$0$$\frac{ 23}{ 128}-\frac{ 11}{ 384}\sqrt{ 33}$ ≈$0.064331-0.146063I$ ≈$0.064331+0.146063I$ ≈$0.115746$$\frac{ 23}{ 128}+\frac{ 11}{ 384}\sqrt{ 33}$ ≈$0.932793$$\infty$
$0$$0$$0$$0$$0$$0$$0$$0$$0$$\frac{ 3}{ 4}$
$1$$1$$0$$1$$1$$1$$1$$1$$1$$1$
$1$$3$$0$$1$$1$$1$$3$$1$$3$$1$
$2$$4$$0$$2$$2$$2$$4$$2$$4$$\frac{ 5}{ 4}$

#### Note:

This is operator "11.13" from ...

12

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 ...

13

New Number: 7.10 |  AESZ:  |  Superseeker: 1 11  |  Hash: b1c277f62ba740f9f7e0371ba53e4194

Degree: 7

$\theta^4-x\left(76\theta^4+80\theta^3+73\theta^2+33\theta+6\right)+x^{2}\left(2209\theta^4+4228\theta^3+4745\theta^2+2726\theta+648\right)-2 3^{2} x^{3}\left(1735\theta^4+4646\theta^3+6099\theta^2+4072\theta+1124\right)+2^{2} 3^{3} x^{4}\left(2085\theta^4+7388\theta^3+11695\theta^2+9140\theta+2844\right)-2^{3} 3^{3} x^{5}(\theta+1)(3707\theta^3+14055\theta^2+20242\theta+10704)+2^{6} 3^{5} x^{6}(\theta+1)(\theta+2)(86\theta^2+285\theta+262)-2^{7} 3^{8} x^{7}(\theta+1)(\theta+2)^2(\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 60, 816, 13104, ...
--> OEIS
Normalized instanton numbers (n0=1): 1, 13/4, 11, 50, 1674/5, ... ; Common denominator:...

#### Discriminant

$-(3z-1)(18z-1)(27z-1)(12z-1)^2(-1+2z)^2$

#### Local exponents

$0$$\frac{ 1}{ 27}$$\frac{ 1}{ 18}$$\frac{ 1}{ 12}$$\frac{ 1}{ 3}$$\frac{ 1}{ 2}$$\infty$
$0$$0$$0$$0$$0$$0$$1$
$0$$1$$1$$1$$1$$\frac{ 1}{ 2}$$2$
$0$$1$$1$$3$$1$$\frac{ 1}{ 2}$$2$
$0$$2$$2$$4$$2$$1$$3$

#### Note:

This is operator "7.10" from ...

14

New Number: 7.7 |  AESZ:  |  Superseeker: 2/3 13/3  |  Hash: c7abb9c42d46f14955f0f23351082bef

Degree: 7

$3^{2} \theta^4-2 3 x\left(88\theta^4+110\theta^3+103\theta^2+48\theta+9\right)+2^{2} x^{2}\left(2923\theta^4+6610\theta^3+8041\theta^2+4908\theta+1206\right)-x^{3}\left(123365\theta^4+374814\theta^3+519741\theta^2+346176\theta+89676\right)+2 x^{4}\left(309657\theta^4+1102938\theta^3+1591157\theta^2+1032920\theta+249740\right)-2^{3} 11 x^{5}(\theta+1)(12897\theta^3+35469\theta^2+31181\theta+8042)-2^{3} 11^{2} x^{6}(\theta+1)(\theta+2)(355\theta^2+1047\theta+806)-2^{4} 11^{3} x^{7}(\theta+1)(\theta+2)^2(\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 56, 636, 8196, ...
--> OEIS
Normalized instanton numbers (n0=1): 2/3, 5/3, 13/3, 59/3, 119, ... ; Common denominator:...

#### Discriminant

$-(11z-1)(4z^2+22z-1)(z^2+11z-1)(-3+22z)^2$

#### Local exponents

$-\frac{ 11}{ 2}-\frac{ 5}{ 2}\sqrt{ 5}$$-\frac{ 11}{ 4}-\frac{ 5}{ 4}\sqrt{ 5}$$0$$-\frac{ 11}{ 4}+\frac{ 5}{ 4}\sqrt{ 5}$$-\frac{ 11}{ 2}+\frac{ 5}{ 2}\sqrt{ 5}$$\frac{ 1}{ 11}$$\frac{ 3}{ 22}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$1$
$1$$1$$0$$1$$1$$1$$1$$2$
$1$$1$$0$$1$$1$$1$$3$$2$
$2$$2$$0$$2$$2$$2$$4$$3$

#### Note:

This is operator "7.7" from ...

15

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 $Q(\sqrt{2})$.

16

New Number: 8.20 |  AESZ: 213  |  Superseeker: 118/17 672  |  Hash: d430b37f4ca641af0b82cbef83547c51

Degree: 8

$17^{2} \theta^4-2 17 x\left(647\theta^4+1240\theta^3+977\theta^2+357\theta+51\right)-2^{2} x^{2}\left(14437\theta^4+89752\theta^3+147734\theta^2+92123\theta+20400\right)+2^{2} 3 x^{3}\left(21538\theta^4+25680\theta^3-41979\theta^2-56151\theta-17442\right)+2^{3} x^{4}\left(51920\theta^4+166384\theta^3-83149\theta^2-217017\theta-79362\right)-2^{4} 3 x^{5}\left(9360\theta^4-26784\theta^3-43813\theta^2-21965\theta-3496\right)-2^{5} 3 x^{6}\left(10160\theta^4-96\theta^3-10535\theta^2-5385\theta-438\right)-2^{8} 3^{2} x^{7}\left(288\theta^4+864\theta^3+1082\theta^2+641\theta+147\right)-2^{11} 3^{2} x^{8}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 162, 6252, 290610, ...
--> OEIS
Normalized instanton numbers (n0=1): 118/17, 873/17, 672, 447987/34, 5358846/17, ... ; Common denominator:...

#### Discriminant

$-(4z+1)(32z^3+40z^2+78z-1)(-17+18z+48z^2)^2$

#### Local exponents

$-\frac{ 3}{ 16}-\frac{ 1}{ 48}\sqrt{ 897}$ ≈$-0.631368-1.433512I$ ≈$-0.631368+1.433512I$$-\frac{ 1}{ 4}$$0$ ≈$0.012736$$-\frac{ 3}{ 16}+\frac{ 1}{ 48}\sqrt{ 897}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$\frac{ 3}{ 4}$
$1$$1$$1$$1$$0$$1$$1$$1$
$3$$1$$1$$1$$0$$1$$3$$1$
$4$$2$$2$$2$$0$$2$$4$$\frac{ 5}{ 4}$

17

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 $Q(\sqrt{-2})$

18

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

Degree: 8

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

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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$Q(\sqrt{6})$.

19

New Number: 8.62 |  AESZ:  |  Superseeker: 127 1566863/3  |  Hash: 4165325308c2b65daacebc1d19717e13

Degree: 8

$\theta^4+x\left(578\theta^4-572\theta^3-359\theta^2-73\theta-6\right)+3^{2} x^{2}\left(4673\theta^4+1892\theta^3+31601\theta^2+11514\theta+1728\right)-2^{3} 3^{4} x^{3}\left(9185\theta^4-134298\theta^3-35420\theta^2-22329\theta-5544\right)+2^{4} 3^{8} x^{4}\left(19051\theta^4+11846\theta^3+114678\theta^2+65939\theta+14290\right)-2^{6} 3^{12} x^{5}\left(7540\theta^4+8068\theta^3-6459\theta^2-7907\theta-2300\right)-2^{6} 3^{16} x^{6}\left(3919\theta^4+27744\theta^3+29957\theta^2+14208\theta+2556\right)+2^{9} 3^{20} 5 x^{7}\left(199\theta^4+590\theta^3+744\theta^2+449\theta+106\right)-2^{12} 3^{24} 5^{2} x^{8}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, -810, -47784, 3354534, ...
--> OEIS
Normalized instanton numbers (n0=1): 127, -14353/2, 1566863/3, -106847355/2, 6507370854, ... ; Common denominator:...

#### Discriminant

$-(81z+1)(419904z^3-22680z^2+79z-1)(-1-288z+29160z^2)^2$

#### Local exponents

$-\frac{ 1}{ 81}$$\frac{ 2}{ 405}-\frac{ 1}{ 1620}\sqrt{ 154}$$0$ ≈$0.001382-0.006675I$ ≈$0.001382+0.006675I$$\frac{ 2}{ 405}+\frac{ 1}{ 1620}\sqrt{ 154}$ ≈$0.051248$$\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 is operator "8.62" from ...

20

New Number: 8.86 |  AESZ:  |  Superseeker: 226/35 3959/7  |  Hash: 815127e123ce989d9ab793a009bb2e6a

Degree: 8

$5^{2} 7^{2} \theta^4-5 7 x\left(3223\theta^4+4862\theta^3+3866\theta^2+1435\theta+210\right)-x^{2}\left(6440-193270\theta-1217171\theta^2-2477628\theta^3-1818051\theta^4\right)-2^{4} 3 x^{3}\left(248985\theta^4+335357\theta^3+239138\theta^2+105280\theta+22400\right)+2^{6} x^{4}\left(618707\theta^4+1107118\theta^3+1179459\theta^2+710680\theta+177284\right)-2^{11} 3 x^{5}\left(12903\theta^4+34738\theta^3+48739\theta^2+33712\theta+8972\right)+2^{15} x^{6}\left(3323\theta^4+12570\theta^3+20137\theta^2+14550\theta+3916\right)-2^{20} x^{7}(\theta+1)(99+295\theta+286\theta^2+88\theta^3)+2^{25} x^{8}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 146, 5280, 229986, ...
--> OEIS
Normalized instanton numbers (n0=1): 226/35, 1599/35, 3959/7, 51101/5, 8052703/35, ... ; Common denominator:...

#### Discriminant

$(1-77z+251z^2-352z^3+512z^4)(32z-5)^2(8z-7)^2$

#### Local exponents

$0$$\frac{ 5}{ 32}$$\frac{ 7}{ 8}$$#ND+#NDI$$\infty$
$0$$0$$0$$0$$1$
$0$$1$$1$$1$$1$
$0$$3$$3$$1$$1$
$0$$4$$4$$2$$1$

#### Note:

This is operator "8.86" from ...

21

New Number: 9.9 |  AESZ:  |  Superseeker: 256/31 28062/31  |  Hash: 924a831431fc249044fe63cfea0eb535

Degree: 9

$31^{2} \theta^4-31 x\left(2836\theta^4+4790\theta^3+3728\theta^2+1333\theta+186\right)-x^{2}\left(1539241\theta^2+1291677\theta+342550-558095\theta^4+131134\theta^3\right)+x^{3}\left(6495560\theta^2+387046\theta^4+6264048\theta^3+558+2100591\theta\right)+x^{4}\left(3388169\theta-7521396\theta^3-5037573\theta^4+2030450-2351908\theta^2\right)-2 x^{5}\left(2014896\theta^4+11047341\theta^3+24693967\theta^2+23008058\theta+7682256\right)+x^{6}\left(37321692\theta+8697364+6817193\theta^4+33832842\theta^3+56561513\theta^2\right)+2 11 x^{7}\left(351229\theta^4+2420534\theta^3+6030705\theta^2+6243956\theta+2275780\right)+2^{2} 11^{2} x^{8}(3667\theta^2+17036\theta+18316)(\theta+1)^2+2^{3} 11^{4} x^{9}(\theta+1)^2(\theta+2)^2$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 178, 7404, 370674, ...
--> OEIS
Normalized instanton numbers (n0=1): 256/31, 1982/31, 28062/31, 591475/31, 15400630/31, ... ; Common denominator:...

#### Discriminant

$(2z+1)(121z^2-86z+1)(z+1)^2(22z^2+147z-31)^2$

#### Local exponents

$-\frac{ 147}{ 44}-\frac{ 1}{ 44}\sqrt{ 24337}$$-1$$-\frac{ 1}{ 2}$$0$$\frac{ 43}{ 121}-\frac{ 24}{ 121}\sqrt{ 3}$$-\frac{ 147}{ 44}+\frac{ 1}{ 44}\sqrt{ 24337}$$\frac{ 43}{ 121}+\frac{ 24}{ 121}\sqrt{ 3}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$1$
$1$$\frac{ 1}{ 2}$$1$$0$$1$$1$$1$$1$
$3$$\frac{ 1}{ 2}$$1$$0$$1$$3$$1$$2$
$4$$1$$2$$0$$2$$4$$2$$2$

#### Note:

This is operator "9.9" from ...

22

New Number: 13.17 |  AESZ:  |  Superseeker: 51/7 4071/7  |  Hash: 1eb2bb810b4c7f191a26886aee350e18

Degree: 13

$5^{2} 7^{2} \theta^4-3 5^{2} 7 x\left(169\theta^4+342\theta^3+269\theta^2+98\theta+14\right)-2 5 x^{2}\left(29068\theta^4+101254\theta^3+142979\theta^2+94430\theta+24780\right)-5 x^{3}\left(72227\theta^4+286050\theta^3+501033\theta^2+425670\theta+139608\right)+2 x^{4}\left(286748\theta^4-779402\theta^3-3422963\theta^2-2684470\theta-681300\right)-x^{5}\left(7490076+19892278\theta+15897011\theta^2-984006\theta^3-2224575\theta^4\right)+2 x^{6}\left(1109623\theta^4+1537878\theta^3-5243929\theta^2-10596978\theta-5189688\right)-2^{2} x^{7}\left(237446\theta^4-1827746\theta^3+1743127\theta^2+3795959\theta+1620252\right)-2^{3} 3^{2} x^{8}\left(58344\theta^4-162618\theta^3-74839\theta^2+120781\theta+86822\right)-2^{2} 3^{2} x^{9}\left(77741\theta^4-159874\theta^3-463443\theta^2-327512\theta-56132\right)+2^{3} 3^{3} x^{10}\left(721\theta^4+12222\theta^3+39317\theta^2+44772\theta+17268\right)-2^{5} 3^{3} x^{11}(\theta+1)(657\theta^3+1363\theta^2+689\theta-222)-2^{5} 3^{3} 13 x^{12}(\theta+2)(\theta+1)(115\theta^2+339\theta+270)-2^{6} 3^{3} 13^{2} x^{13}(\theta+1)(\theta+2)^2(\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 156, 5796, 259296, ...
--> OEIS
Normalized instanton numbers (n0=1): 51/7, 1552/35, 4071/7, 378248/35, 1721920/7, ... ; Common denominator:...

#### Discriminant



No data for singularities

#### Note:

This is operator "13.17" from ...