### Summary

You searched for: sol=18

1

New Number: 2.10 |  AESZ: 70  |  Superseeker: 27 18089  |  Hash: 3d2adae6eaf26a56c76b8b67d92cc5df

Degree: 2

$\theta^4-3 x(3\theta+1)(3\theta+2)(10\theta^2+10\theta+3)+3^{4} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 1350, 156240, 22141350, ...
--> OEIS
Normalized instanton numbers (n0=1): 27, 432, 18089, 997785, 68438142, ... ; Common denominator:...

#### Discriminant

$(243z-1)(27z-1)$

#### Local exponents

$0$$\frac{ 1}{ 243}$$\frac{ 1}{ 27}$$\infty$
$0$$0$$0$$\frac{ 1}{ 3}$
$0$$1$$1$$\frac{ 2}{ 3}$
$0$$1$$1$$\frac{ 4}{ 3}$
$0$$2$$2$$\frac{ 5}{ 3}$

#### Note:

Hadamard product $B\ast c$.

2

New Number: 2.21 |  AESZ: 134  |  Superseeker: 18 -5177  |  Hash: cc6d92c4b8a8dadb92b447c54e3a2a2f

Degree: 2

$\theta^4-3^{2} x(3\theta+1)(3\theta+2)(3\theta^2+3\theta+1)+3^{5} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 810, 35280, 311850, ...
--> OEIS
Normalized instanton numbers (n0=1): 18, -207/2, -5177, -155979, -923301, ... ; Common denominator:...

#### Discriminant

$1-243z+19683z^2$

#### Local exponents

$0$$\frac{ 1}{ 162}-\frac{ 1}{ 486}\sqrt{ 3}I$$\frac{ 1}{ 162}+\frac{ 1}{ 486}\sqrt{ 3}I$$\infty$
$0$$0$$0$$\frac{ 1}{ 3}$
$0$$1$$1$$\frac{ 2}{ 3}$
$0$$1$$1$$\frac{ 4}{ 3}$
$0$$2$$2$$\frac{ 5}{ 3}$

#### Note:

Hadamard product $B\ast f$

3

New Number: 2.6 |  AESZ: 24  |  Superseeker: 36 41421  |  Hash: 5e8f8f32b5e99693a2956e1240b9fdff

Degree: 2

$\theta^4-3 x(3\theta+1)(3\theta+2)(11\theta^2+11\theta+3)-3^{2} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 1710, 246960, 43347150, ...
--> OEIS
Normalized instanton numbers (n0=1): 36, 837, 41421, 2992851, 266362506, ... ; Common denominator:...

#### Discriminant

$1-297z-729z^2$

#### Local exponents

$-\frac{ 11}{ 54}-\frac{ 5}{ 54}\sqrt{ 5}$$0$$-\frac{ 11}{ 54}+\frac{ 5}{ 54}\sqrt{ 5}$$\infty$
$0$$0$$0$$\frac{ 1}{ 3}$
$1$$0$$1$$\frac{ 2}{ 3}$
$1$$0$$1$$\frac{ 4}{ 3}$
$2$$0$$2$$\frac{ 5}{ 3}$

#### Note:

Related to 7.19, 8.18
This operator corresponds to $(Grass(2,5)\vert 1,1,3)_{-150}$ from arXiv:0802.2908

4

New Number: 3.10 |  AESZ: ~103  |  Superseeker: 10 664  |  Hash: 9239615e8ac132ca232c13367a39ae3b

Degree: 3

$\theta^4-2 x\left(86\theta^4+172\theta^3+143\theta^2+57\theta+9\right)+2^{2} 3^{2} x^{2}(\theta+1)^2(236\theta^2+472\theta+187)-2^{4} 3^{4} 5^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 630, 28980, 1593270, ...
--> OEIS
Normalized instanton numbers (n0=1): 10, 24, 664, 9088, 234388, ... ; Common denominator:...

#### Discriminant

$-(100z-1)(-1+36z)^2$

#### Local exponents

$0$$\frac{ 1}{ 100}$$\frac{ 1}{ 36}$$\infty$
$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$\frac{ 1}{ 2}$$1$
$0$$1$$\frac{ 1}{ 2}$$2$
$0$$2$$1$$\frac{ 5}{ 2}$

#### Note:

Operator equivalent to AESZ 103 =$c \ast c$.

5

New Number: 5.109 |  AESZ: 373  |  Superseeker: 50 68472  |  Hash: 8b0fbfc0016c3fb02fd42d4ff919e0f8

Degree: 5

$\theta^4-2 x\left(190\theta^4+308\theta^3+227\theta^2+73\theta+9\right)+2^{2} x^{2}\left(4780\theta^4+6304\theta^3+2395\theta^2+642\theta+135\right)-2^{4} 3 x^{3}\left(6700\theta^4+8472\theta^3+7607\theta^2+3615\theta+648\right)+2^{7} 3^{2} x^{4}(2\theta+1)(760\theta^3+1464\theta^2+1211\theta+375)-2^{10} 3^{6} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 1782, 276660, 52396470, ...
--> OEIS
Normalized instanton numbers (n0=1): 50, 1299, 68472, 5536032, 555252324, ... ; Common denominator:...

#### Discriminant

$-(-1+324z)(24z-1)^2(4z-1)^2$

#### Local exponents

$0$$\frac{ 1}{ 324}$$\frac{ 1}{ 24}$$\frac{ 1}{ 4}$$\infty$
$0$$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$\frac{ 1}{ 2}$$1$
$0$$1$$3$$\frac{ 1}{ 2}$$1$
$0$$2$$4$$1$$\frac{ 3}{ 2}$

#### Note:

This is operator "5.109" from ...

6

New Number: 5.82 |  AESZ: 313  |  Superseeker: 45 43531  |  Hash: f8bfe82988e14680bdb775a3ce956216

Degree: 5

$\theta^4-x(\theta+1)(285\theta^3+321\theta^2+128\theta+18)-2 x^{2}\left(1640\theta^4+1322\theta^3-1337\theta^2-1178\theta-240\right)-2^{2} 3^{2} x^{3}\left(213\theta^4-256\theta^3-286\theta^2-80\theta-5\right)+2^{3} 3^{3} x^{4}(2\theta+1)(22\theta^3+37\theta^2+24\theta+6)+2^{4} 3^{3} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 1662, 236340, 40943070, ...
--> OEIS
Normalized instanton numbers (n0=1): 45, 845, 43531, 3091112, 273471538, ... ; Common denominator:...

#### Discriminant

$(z-1)(48z^2+296z-1)(6z+1)^2$

#### Local exponents

$-\frac{ 37}{ 12}-\frac{ 7}{ 6}\sqrt{ 7}$$-\frac{ 1}{ 6}$$0$$-\frac{ 37}{ 12}+\frac{ 7}{ 6}\sqrt{ 7}$$1$$\infty$
$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$1$$0$$1$$1$$1$
$1$$3$$0$$1$$1$$1$
$2$$4$$0$$2$$2$$\frac{ 3}{ 2}$

#### Note:

This is operator "5.82" from ...

7

New Number: 11.2 |  AESZ:  |  Superseeker: 136/97 1768/97  |  Hash: 940a6a9fb87fe9b9613bd73b990374c1

Degree: 11

$97^{2} \theta^4+97 x\theta(1727\theta^3-2018\theta^2-1300\theta-291)-x^{2}\left(1652135\theta^4+13428812\theta^3+16174393\theta^2+10216234\theta+2709792\right)-3 x^{3}\left(27251145\theta^4+121375398\theta^3+189546499\theta^2+147705198\theta+46000116\right)-2 x^{4}\left(587751431\theta^4+2711697232\theta^3+5003189285\theta^2+4434707760\theta+1524637512\right)-x^{5}\left(9726250397\theta^4+50507429234\theta^3+106108023451\theta^2+103964102350\theta+38537290992\right)-2 3 x^{6}\left(8793822649\theta^4+52062405804\theta^3+122175610025\theta^2+130254629814\theta+51340027968\right)-2^{2} 3^{2} x^{7}\left(5429262053\theta^4+36477756530\theta^3+94431307279\theta^2+108363704338\theta+44982230808\right)-2^{4} 3^{2} x^{8}(\theta+1)(3432647479\theta^3+22487363787\theta^2+50808614711\theta+38959393614)-2^{4} 3^{3} x^{9}(\theta+1)(\theta+2)(1903493629\theta^2+10262864555\theta+14314039440)-2^{5} 3^{4} 13^{2} x^{10}(\theta+3)(\theta+2)(\theta+1)(1862987\theta+5992902)-2^{6} 3^{3} 13^{4} 7457 x^{11}(\theta+1)(\theta+2)(\theta+3)(\theta+4)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 0, 18, 168, 2430, ...
--> OEIS
Normalized instanton numbers (n0=1): 136/97, 292/97, 1768/97, 10128/97, 83387/97, ... ; Common denominator:...

#### Discriminant

$-(12z^2+6z+1)(7457z^5+6100z^4+1929z^3+257z^2+7z-1)(97+912z+2028z^2)^2$

#### Local exponents

$-\frac{ 38}{ 169}-\frac{ 1}{ 1014}\sqrt{ 2805}$$-\frac{ 1}{ 4}-\frac{ 1}{ 12}\sqrt{ 3}I$$-\frac{ 1}{ 4}+\frac{ 1}{ 12}\sqrt{ 3}I$$-\frac{ 38}{ 169}+\frac{ 1}{ 1014}\sqrt{ 2805}$$0$$#ND+#NDI$$\infty$
$0$$0$$0$$0$$0$$0$$1$
$1$$1$$1$$1$$0$$1$$2$
$3$$1$$1$$3$$0$$1$$3$
$4$$2$$2$$4$$0$$2$$4$

#### Note:

This is operator "11.2" from ...

8

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 ...

9

New Number: 6.10 |  AESZ:  |  Superseeker: 23 16723  |  Hash: 23025d094839fb9d8e76076bd9a0bfa7

Degree: 6

$\theta^4-x\left(254\theta^4+508\theta^3+391\theta^2+137\theta+18\right)+x^{2}\left(4657\theta^4+18628\theta^3+27265\theta^2+17274\theta+3672\right)-2^{2} 3 x^{3}\left(2920\theta^4+17520\theta^3+36833\theta^2+31659\theta+8235\right)+2^{3} 3^{4} x^{4}\left(204\theta^4+1632\theta^3+4449\theta^2+4740\theta+1400\right)-2^{4} 3^{5} x^{5}(16\theta^2+80\theta+35)(2\theta+5)^2+2^{4} 3^{6} x^{6}(2\theta+11)(2\theta+7)(2\theta+5)(2\theta+1)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 1242, 138420, 18954810, ...
--> OEIS
Normalized instanton numbers (n0=1): 23, 462, 16723, 923487, 61874817, ... ; Common denominator:...

#### Discriminant

$(3z-1)(3888z^3-1944z^2+243z-1)(4z-1)^2$

#### Local exponents

$0$ ≈$0.004259$ ≈$0.215449$$\frac{ 1}{ 4}$ ≈$0.280292$$\frac{ 1}{ 3}$$\infty$
$0$$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$\frac{ 1}{ 2}$$1$$1$$\frac{ 5}{ 2}$
$0$$1$$1$$\frac{ 1}{ 2}$$1$$1$$\frac{ 7}{ 2}$
$0$$2$$2$$1$$2$$2$$\frac{ 11}{ 2}$

#### Note:

This is operator "6.10" from ...

10

New Number: 6.18 |  AESZ:  |  Superseeker: 3 64  |  Hash: b127e33287ed87a366c178fc4678cdc4

Degree: 6

$\theta^4-x\left(18+94\theta+199\theta^2+210\theta^3+105\theta^4\right)+2 x^{2}\left(2095\theta^4+8380\theta^3+13298\theta^2+9836\theta+2850\right)-2^{2} 3^{2} x^{3}\left(2310\theta^4+13860\theta^3+30739\theta^2+29847\theta+10763\right)+2^{3} 3^{3} x^{4}\left(4044\theta^4+32352\theta^3+91997\theta^2+109172\theta+45693\right)-2^{4} 3^{3} 5^{2} 7 x^{5}(\theta+4)(\theta+1)(61\theta^2+305\theta+345)+2^{5} 3^{5} 5^{2} 7^{2} x^{6}(\theta+5)(\theta+4)(\theta+2)(\theta+1)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 348, 7320, 168840, ...
--> OEIS
Normalized instanton numbers (n0=1): 3, -4, 64, -253, 4292, ... ; Common denominator:...

#### Discriminant

$(6z-1)(15z-1)(14z-1)(42z-1)(18z-1)(10z-1)$

#### Local exponents

$0$$\frac{ 1}{ 42}$$\frac{ 1}{ 18}$$\frac{ 1}{ 15}$$\frac{ 1}{ 14}$$\frac{ 1}{ 10}$$\frac{ 1}{ 6}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$1$
$0$$1$$1$$1$$1$$1$$1$$2$
$0$$1$$1$$1$$1$$1$$1$$4$
$0$$2$$2$$2$$2$$2$$2$$5$

#### Note:

This is operator "6.18" from ...

11

New Number: 6.7 |  AESZ:  |  Superseeker: -9 217/3  |  Hash: 9492f991c909a6774f5546668ff53b6a

Degree: 6

$\theta^4-3 x\left(42\theta^4+84\theta^3+77\theta^2+35\theta+6\right)+3^{3} x^{2}\left(291\theta^4+1164\theta^3+1747\theta^2+1166\theta+264\right)-2^{2} 3^{5} x^{3}\left(360\theta^4+2160\theta^3+4553\theta^2+3939\theta+1035\right)+2^{3} 3^{8} x^{4}\left(204\theta^4+1632\theta^3+4449\theta^2+4740\theta+1400\right)-2^{4} 3^{11} x^{5}(16\theta^2+80\theta+35)(2\theta+5)^2+2^{4} 3^{14} x^{6}(2\theta+11)(2\theta+7)(2\theta+5)(2\theta+1)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 378, 8820, 266490, ...
--> OEIS
Normalized instanton numbers (n0=1): -9, 18, 217/3, -9, -146079, ... ; Common denominator:...

#### Discriminant

$(27z-1)(34992z^3-1944z^2+27z-1)(-1+36z)^2$

#### Local exponents

$0$ ≈$0.002095-0.023494I$ ≈$0.002095+0.023494I$$\frac{ 1}{ 36}$$\frac{ 1}{ 27}$ ≈$0.051365$$\infty$
$0$$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$\frac{ 1}{ 2}$$1$$1$$\frac{ 5}{ 2}$
$0$$1$$1$$\frac{ 1}{ 2}$$1$$1$$\frac{ 7}{ 2}$
$0$$2$$2$$1$$2$$2$$\frac{ 11}{ 2}$

#### Note:

This is operator "6.7" from ...

12

New Number: 8.12 |  AESZ: 175  |  Superseeker: 17 1387  |  Hash: f6db11b5e593983f455489d5bb1003c5

Degree: 8

$\theta^4-x(10\theta^2+10\theta+3)(17\theta^2+17\theta+6)+3^{4} x^{2}\left(89\theta^4+452\theta^3+633\theta^2+362\theta+80\right)+2^{3} 3^{4} x^{3}\left(170\theta^4-1020\theta^3-3119\theta^2-2373\theta-648\right)-2^{4} 3^{8} x^{4}\left(97\theta^4+194\theta^3-238\theta^2-335\theta-114\right)+2^{6} 3^{8} x^{5}\left(170\theta^4+1700\theta^3+961\theta^2-125\theta-204\right)+2^{6} 3^{12} x^{6}\left(89\theta^4-96\theta^3-189\theta^2-96\theta-12\right)-2^{9} 3^{12} x^{7}(10\theta^2+10\theta+3)(17\theta^2+17\theta+6)+2^{12} 3^{16} x^{8}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 630, 29016, 1529766, ...
--> OEIS
Normalized instanton numbers (n0=1): 17, -299/4, 1387, -47623/2, 500282, ... ; Common denominator:...

#### Discriminant

$(81z-1)(8z-1)(72z-1)(9z-1)(-1+648z^2)^2$

#### Local exponents

$-\frac{ 1}{ 36}\sqrt{ 2}$$0$$\frac{ 1}{ 81}$$\frac{ 1}{ 72}$$\frac{ 1}{ 36}\sqrt{ 2}$$\frac{ 1}{ 9}$$\frac{ 1}{ 8}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$1$
$1$$0$$1$$1$$1$$1$$1$$1$
$3$$0$$1$$1$$3$$1$$1$$1$
$4$$0$$2$$2$$4$$2$$2$$1$

#### Note:

Hadamard product $c \ast g$. 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 $Q(\sqrt{?})$.

13

New Number: 8.15 |  AESZ: 178  |  Superseeker: 18 9799/3  |  Hash: e748913f322a008ae5c350f96f1cd860

Degree: 8

$\theta^4-3 x(3\theta^2+3\theta+1)(17\theta^2+17\theta+6)+3^{3} x^{2}\left(217\theta^4+1732\theta^3+2441\theta^2+1418\theta+336\right)+2^{3} 3^{6} x^{3}\left(51\theta^4-306\theta^3-934\theta^2-717\theta-204\right)-2^{4} 3^{8} x^{4}\left(289\theta^4+578\theta^3-1310\theta^2-1599\theta-570\right)+2^{6} 3^{11} x^{5}\left(51\theta^4+510\theta^3+290\theta^2-29\theta-64\right)+2^{6} 3^{13} x^{6}\left(217\theta^4-864\theta^3-1453\theta^2-864\theta-156\right)-2^{9} 3^{16} x^{7}(3\theta^2+3\theta+1)(17\theta^2+17\theta+6)+2^{12} 3^{20} x^{8}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 378, 6552, 21546, ...
--> OEIS
Normalized instanton numbers (n0=1): 18, -423/2, 9799/3, -150003/2, 1914237, ... ; Common denominator:...

#### Discriminant

$(1728z^2-72z+1)(2187z^2-81z+1)(-1+1944z^2)^2$

#### Local exponents

$-\frac{ 1}{ 108}\sqrt{ 6}$$0$$\frac{ 1}{ 54}-\frac{ 1}{ 162}\sqrt{ 3}I$$\frac{ 1}{ 54}+\frac{ 1}{ 162}\sqrt{ 3}I$$\frac{ 1}{ 48}-\frac{ 1}{ 144}\sqrt{ 3}I$$\frac{ 1}{ 48}+\frac{ 1}{ 144}\sqrt{ 3}I$$\frac{ 1}{ 108}\sqrt{ 6}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$1$
$1$$0$$1$$1$$1$$1$$1$$1$
$3$$0$$1$$1$$1$$1$$3$$1$
$4$$0$$2$$2$$2$$2$$4$$1$

#### Note:

Hadamard product $d \ast g$. This operator has a second MUM-point at infinity. It can be reduced to an operator of degree 4 with a single MUM-point defined over $Q(\sqrt{?})$.

14

New Number: 8.30 |  AESZ: 314  |  Superseeker: 229/4 297111/4  |  Hash: 893692ba7eb3effcbc0c3b48d405456a

Degree: 8

$2^{4} \theta^4-2^{2} x\left(1282\theta^4+2618\theta^3+1909\theta^2+600\theta+72\right)-3^{2} x^{2}\left(9503\theta^4+26810\theta^3+31755\theta^2+15944\theta+2936\right)+3^{4} x^{3}\left(15627\theta^4-18288\theta^3-91412\theta^2-53256\theta-9688\right)+2 3^{6} x^{4}\left(15106\theta^4+20300\theta^3-20421\theta^2-23443\theta-5907\right)-2^{2} 3^{8} x^{5}\left(2072\theta^4-18256\theta^3-2563\theta^2+4626\theta+1495\right)-2^{2} 3^{10} x^{6}\left(6204\theta^4+360\theta^3-281\theta^2+1017\theta+434\right)-2^{5} 3^{12} x^{7}(2\theta+1)(100\theta^3+162\theta^2+95\theta+21)+2^{8} 3^{14} x^{8}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 1926, 310860, 61060230, ...
--> OEIS
Normalized instanton numbers (n0=1): 229/4, 1293, 297111/4, 6150238, 2540085295/4, ... ; Common denominator:...

#### Discriminant

$(z-1)(11664z^3+3888z^2+324z-1)(-4-9z+648z^2)^2$

#### Local exponents

≈$-0.168156-0.022431I$ ≈$-0.168156+0.022431I$$\frac{ 1}{ 144}-\frac{ 1}{ 144}\sqrt{ 129}$$0$$\frac{ 1}{ 18}2^(\frac{ 1}{ 3})+\frac{ 1}{ 36}2^(\frac{ 2}{ 3})-\frac{ 1}{ 9}$$\frac{ 1}{ 144}+\frac{ 1}{ 144}\sqrt{ 129}$$1$$\infty$
$0$$0$$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$1$$1$$0$$1$$1$$1$$1$
$1$$1$$3$$0$$1$$3$$1$$1$
$2$$2$$4$$0$$2$$4$$2$$\frac{ 3}{ 2}$

#### Note:

This is operator "8.30" from ...

15

New Number: 8.9 |  AESZ: 174  |  Superseeker: 16 -13  |  Hash: 3f987b46d9ebf201eeead1a885b78e66

Degree: 8

$\theta^4-x(11\theta^2+11\theta+3)(17\theta^2+17\theta+6)+x^{2}\left(8711\theta^4+33980\theta^3+47095\theta^2+26230\theta+5232\right)-2^{3} 3^{2} x^{3}\left(187\theta^4-1122\theta^3-3436\theta^2-2595\theta-684\right)+2^{4} 3^{2} x^{4}\left(8639\theta^4+17278\theta^3-11650\theta^2-20289\theta-6102\right)+2^{6} 3^{4} x^{5}\left(187\theta^4+1870\theta^3+1052\theta^2-163\theta-216\right)+2^{6} 3^{4} x^{6}\left(8711\theta^4+864\theta^3-2579\theta^2+864\theta+828\right)+2^{9} 3^{6} x^{7}(11\theta^2+11\theta+3)(17\theta^2+17\theta+6)+2^{12} 3^{8} x^{8}\left((\theta+1)^4\right)$

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Coefficients of the holomorphic solution: 1, 18, 798, 45864, 2994894, ...
--> OEIS
Normalized instanton numbers (n0=1): 16, 7/2, -13, 11663/2, -26414, ... ; Common denominator:...

#### Discriminant

$(81z^2+99z-1)(64z^2+88z-1)(1+72z^2)^2$

#### Local exponents

$-\frac{ 11}{ 16}-\frac{ 5}{ 16}\sqrt{ 5}$$-\frac{ 11}{ 18}-\frac{ 5}{ 18}\sqrt{ 5}$$0-\frac{ 1}{ 12}\sqrt{ 2}I$$0$$0+\frac{ 1}{ 12}\sqrt{ 2}I$$-\frac{ 11}{ 18}+\frac{ 5}{ 18}\sqrt{ 5}$$-\frac{ 11}{ 16}+\frac{ 5}{ 16}\sqrt{ 5}$$\infty$
$0$$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 $b \ast g$. 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 $Q(\sqrt{?})$.

16

New Number: 9.8 |  AESZ:  |  Superseeker: 6/17 688/17  |  Hash: 0574d9effd306eb6c9288752b7670904

Degree: 9

$17^{2} \theta^4-2 17 x\left(164\theta^4-164\theta^3-167\theta^2-85\theta-17\right)-2^{2} x^{2}\left(35300\theta^4+95864\theta^3+121575\theta^2+70856\theta+16235\right)+2^{2} x^{3}\left(427984\theta^4-277824\theta^3-1460293\theta^2-1490475\theta-492694\right)+2^{4} x^{4}\left(2088512\theta^4+6692704\theta^3+7319011\theta^2+3820745\theta+794302\right)-2^{6} x^{5}\left(1379872\theta^4-6413120\theta^3-11843583\theta^2-9110135\theta-2589134\right)-2^{8} x^{6}\left(13237904\theta^4+37140384\theta^3+64254239\theta^2+57084594\theta+19379105\right)-2^{10} 3^{2} 5 x^{7}\left(255072\theta^4+803200\theta^3+1114259\theta^2+709496\theta+167515\right)+2^{12} 3^{3} 5^{2} 7 x^{8}(2224\theta^2+11008\theta+12225)(\theta+1)^2+2^{18} 3^{3} 5^{4} 7^{2} x^{9}(\theta+1)^2(\theta+2)^2$

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Coefficients of the holomorphic solution: 1, -2, 18, -20, 1330, ...
--> OEIS
Normalized instanton numbers (n0=1): 6/17, 83/17, 688/17, 7350/17, 5150, ... ; Common denominator:...

#### Discriminant

$(4z-1)(12z+1)(1600z^3+272z^2+8z-1)(-17+164z+1680z^2)^2$

#### Local exponents

$-\frac{ 41}{ 840}-\frac{ 1}{ 840}\sqrt{ 8821}$ ≈$-0.106819-0.053966I$ ≈$-0.106819+0.053966I$$-\frac{ 1}{ 12}$$0$ ≈$0.043637$$-\frac{ 41}{ 840}+\frac{ 1}{ 840}\sqrt{ 8821}$$\frac{ 1}{ 4}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$0$$1$
$1$$1$$1$$1$$0$$1$$1$$1$$1$
$3$$1$$1$$1$$0$$1$$3$$1$$2$
$4$$2$$2$$2$$0$$2$$4$$2$$2$

#### Note:

This is operator "9.8" from ...