### Summary

You searched for: sol=324

1

New Number: 2.20 |  AESZ: 133  |  Superseeker: 12 -3284/3  |  Hash: 4c9628f7dd48f4e9e6ec75303e557389

Degree: 2

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

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Coefficients of the holomorphic solution: 1, 12, 324, 8400, 44100, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, -42, -3284/3, -20538, -103776, ... ; Common denominator:...

#### Discriminant

$1-144z+6912z^2$

#### Local exponents

$0$$\frac{ 1}{ 96}-\frac{ 1}{ 288}\sqrt{ 3}I$$\frac{ 1}{ 96}+\frac{ 1}{ 288}\sqrt{ 3}I$$\infty$
$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$\frac{ 1}{ 2}$
$0$$1$$1$$\frac{ 3}{ 2}$
$0$$2$$2$$\frac{ 3}{ 2}$

#### Note:

Explicit solution not yet verified

2

New Number: 2.65 |  AESZ: 183  |  Superseeker: -4 -556/9  |  Hash: 04a3788c3f9ed53281ae824deb33d833

Degree: 2

$\theta^4+2^{2} x(2\theta+1)^2(7\theta^2+7\theta+3)+2^{4} 3 x^{2}(2\theta+1)(4\theta+3)(4\theta+5)(2\theta+3)$

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Coefficients of the holomorphic solution: 1, -12, 324, -11280, 447300, ...
--> OEIS
Normalized instanton numbers (n0=1): -4, 8, -556/9, 624, -8928, ... ; Common denominator:...

#### Discriminant

$(48z+1)(64z+1)$

#### Local exponents

$-\frac{ 1}{ 48}$$-\frac{ 1}{ 64}$$0$$\infty$
$0$$0$$0$$\frac{ 1}{ 2}$
$1$$1$$0$$\frac{ 3}{ 4}$
$1$$1$$0$$\frac{ 5}{ 4}$
$2$$2$$0$$\frac{ 3}{ 2}$

#### Note:

This is operator "2.65" from ...

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.

4

New Number: 4.72 |  AESZ: 361  |  Superseeker: 20 -119332/9  |  Hash: f55eaa640956f064f5230c04d8173d60

Degree: 4

$\theta^4-2^{2} x\left(80\theta^4+88\theta^3+67\theta^2+23\theta+3\right)+2^{4} 3 x^{2}\left(928\theta^4+2080\theta^3+2176\theta^2+972\theta+153\right)-2^{10} 3^{2} x^{3}\left(272\theta^4+648\theta^3+511\theta^2+162\theta+18\right)+2^{12} 3^{6} x^{4}\left((2\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 324, -6000, -2778300, ...
--> OEIS
Normalized instanton numbers (n0=1): 20, -139, -119332/9, -462222, -2113440, ... ; Common denominator:...

#### Discriminant

$(20736z^2-224z+1)(-1+48z)^2$

#### Local exponents

$0$$s_1$$s_2$$\frac{ 7}{ 1296}-\frac{ 1}{ 324}\sqrt{ 2}I$$\frac{ 7}{ 1296}+\frac{ 1}{ 324}\sqrt{ 2}I$$\frac{ 1}{ 48}$$\infty$
$0$$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$1$$1$$1$$\frac{ 1}{ 2}$
$0$$1$$1$$1$$1$$3$$\frac{ 1}{ 2}$
$0$$2$$2$$2$$2$$4$$\frac{ 1}{ 2}$

#### Note:

Sporadic Operator. There is a second MUM-point hiding at infinity, corresponding to Operator AESZ 362/4.73

5

New Number: 5.51 |  AESZ: 250  |  Superseeker: 308/23 70799/23  |  Hash: 9c19794a84073d1c6dfd11c8a7c9a740

Degree: 5

$23^{2} \theta^4-23 x\left(3271\theta^4+5078\theta^3+3896\theta^2+1357\theta+184\right)+x^{2}\left(1357863\theta^4+999924\theta^3-787393\theta^2-850862\theta-205712\right)-2^{3} x^{3}\left(775799\theta^4-272481\theta^3-218821\theta^2+176709\theta+100234\right)-2^{4} 61 x^{4}\left(1005\theta^4-15654\theta^3-36317\theta^2-27938\theta-7304\right)-2^{9} 61^{2} x^{5}(4\theta+3)(\theta+1)^2(4\theta+5)$

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Coefficients of the holomorphic solution: 1, 8, 324, 19304, 1388260, ...
--> OEIS
Normalized instanton numbers (n0=1): 308/23, 3526/23, 70799/23, 2148684/23, 81402822/23, ... ; Common denominator:...

#### Discriminant

$-(512z^3+113z^2+121z-1)(-23+244z)^2$

#### Local exponents

≈$-0.114451-0.474453I$ ≈$-0.114451+0.474453I$$0$ ≈$0.008199$$\frac{ 23}{ 244}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 3}{ 4}$
$1$$1$$0$$1$$1$$1$
$1$$1$$0$$1$$3$$1$
$2$$2$$0$$2$$4$$\frac{ 5}{ 4}$

#### Note:

This is operator "5.51" from ...

6

New Number: 5.66 |  AESZ: 274  |  Superseeker: 49/5 6032/15  |  Hash: 729d44a3b7b561b49603f26a25d26069

Degree: 5

$5^{2} \theta^4-5 x\left(757\theta^4+1298\theta^3+1049\theta^2+400\theta+60\right)+2^{2} 3^{2} x^{2}\left(5456\theta^4+17498\theta^3+22121\theta^2+11940\theta+2340\right)-2^{2} 3^{4} x^{3}\left(15128\theta^4+68040\theta^3+112171\theta^2+73845\theta+16380\right)+2^{4} 3^{8} x^{4}(2\theta+1)(216\theta^3+864\theta^2+1015\theta+356)-2^{6} 3^{10} x^{5}(2\theta+1)(3\theta+2)(3\theta+4)(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 324, 12000, 548100, ...
--> OEIS
Normalized instanton numbers (n0=1): 49/5, -68/5, 6032/15, 36276/5, 350082/5, ... ; Common denominator:...

#### Discriminant

$-(81z-1)(1296z^2-56z+1)(-5+36z)^2$

#### Local exponents

$0$$\frac{ 1}{ 81}$$\frac{ 7}{ 324}-\frac{ 1}{ 81}\sqrt{ 2}I$$\frac{ 7}{ 324}+\frac{ 1}{ 81}\sqrt{ 2}I$$\frac{ 5}{ 36}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$1$$1$$\frac{ 2}{ 3}$
$0$$1$$1$$1$$3$$\frac{ 4}{ 3}$
$0$$2$$2$$2$$4$$\frac{ 3}{ 2}$

#### Note:

This is operator "5.66" from ...

7

New Number: 6.5 |  AESZ:  |  Superseeker: -11 -3422/3  |  Hash: 6a4aeb5833b7673c962d5598842d3f2c

Degree: 6

$\theta^4-x\left(12+64\theta+125\theta^2+122\theta^3+61\theta^4\right)-2^{3} x^{2}\left(193\theta^4+772\theta^3+1033\theta^2+522\theta+72\right)+2^{9} 3 x^{3}\left(146\theta^4+876\theta^3+1838\theta^2+1572\theta+405\right)-2^{12} 3^{2} x^{4}\left(204\theta^4+1632\theta^3+4449\theta^2+4740\theta+1400\right)+2^{16} 3^{3} x^{5}(16\theta^2+80\theta+35)(2\theta+5)^2-2^{19} 3^{4} x^{6}(2\theta+11)(2\theta+7)(2\theta+5)(2\theta+1)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 324, 5760, 215460, ...
--> OEIS
Normalized instanton numbers (n0=1): -11, 68, -3422/3, 30735, -1014993, ... ; Common denominator:...

#### Discriminant

$-(24z-1)(27648z^3-1728z^2+27z+1)(-1+32z)^2$

#### Local exponents

≈$-0.016119$$0$$\frac{ 1}{ 32}$ ≈$0.03931-0.026431I$ ≈$0.03931+0.026431I$$\frac{ 1}{ 24}$$\infty$
$0$$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$0$$\frac{ 1}{ 2}$$1$$1$$1$$\frac{ 5}{ 2}$
$1$$0$$\frac{ 1}{ 2}$$1$$1$$1$$\frac{ 7}{ 2}$
$2$$0$$1$$2$$2$$2$$\frac{ 11}{ 2}$

#### Note:

This is operator "6.5" from ...

8

New Number: 8.84 |  AESZ:  |  Superseeker: 1/5 224/5  |  Hash: 258fab6f0a4f132fe597fc6f30e54eea

Degree: 8

$5^{2} \theta^4+5 x\theta^2(-1-2\theta+107\theta^2)+2^{2} x^{2}\left(2174\theta^4+5942\theta^3+8569\theta^2+5200\theta+1200\right)+2^{2} 3^{2} x^{3}\left(308\theta^4-4248\theta^3-17051\theta^2-16785\theta-5280\right)-2^{4} 3^{2} x^{4}\left(7060\theta^4+39500\theta^3+69820\theta^2+52851\theta+14688\right)-2^{6} 3^{4} x^{5}\left(881\theta^4+3974\theta^3+8648\theta^2+7983\theta+2581\right)+2^{7} 3^{4} x^{6}\left(1192\theta^4+2376\theta^3-1132\theta^2-4185\theta-1926\right)+2^{8} 3^{6} x^{7}\left(68\theta^4+568\theta^3+1095\theta^2+811\theta+210\right)-2^{12} 3^{8} x^{8}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 0, -12, 144, 324, ...
--> OEIS
Normalized instanton numbers (n0=1): 1/5, -6, 224/5, -448/5, -4334/5, ... ; Common denominator:...

#### Discriminant

$-(9z-1)(576z^3+368z^2+16z+1)(-5-36z+72z^2)^2$

#### Local exponents

≈$-0.597246$$\frac{ 1}{ 4}-\frac{ 1}{ 12}\sqrt{ 19}$ ≈$-0.020821-0.049733I$ ≈$-0.020821+0.049733I$$0$$\frac{ 1}{ 9}$$\frac{ 1}{ 4}+\frac{ 1}{ 12}\sqrt{ 19}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$1$
$1$$1$$1$$1$$0$$1$$1$$1$
$1$$3$$1$$1$$0$$1$$3$$1$
$2$$4$$2$$2$$0$$2$$4$$1$

#### Note:

This is operator "8.84" from ...