### Summary

You searched for: sol=684

1

New Number: 2.5 |  AESZ: 25  |  Superseeker: 20 8220  |  Hash: 93279abcbeeade30c29508de7784e582

Degree: 2

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

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Coefficients of the holomorphic solution: 1, 12, 684, 58800, 6129900, ...
--> OEIS
Normalized instanton numbers (n0=1): 20, 277, 8220, 352994, 18651536, ... ; Common denominator:...

#### Discriminant

$1-176z-256z^2$

#### Local exponents

$-\frac{ 11}{ 32}-\frac{ 5}{ 32}\sqrt{ 5}$$0$$-\frac{ 11}{ 32}+\frac{ 5}{ 32}\sqrt{ 5}$$\infty$
$0$$0$$0$$\frac{ 1}{ 2}$
$1$$0$$1$$\frac{ 1}{ 2}$
$1$$0$$1$$\frac{ 3}{ 2}$
$2$$0$$2$$\frac{ 3}{ 2}$

#### Note:

Hadamard product $A\ast b$

A-incarnation: X(1,2,2) in G(2,5)

2

New Number: 5.50 |  AESZ: 249  |  Superseeker: -44/5 -596  |  Hash: 85592af20bbb190e37428e945664c2f3

Degree: 5

$5^{2} \theta^4+2^{2} 5 x\left(148\theta^4+392\theta^3+341\theta^2+145\theta+25\right)+2^{4} x^{2}\left(4096\theta^4+32128\theta^3+57016\theta^2+37920\theta+9175\right)-2^{8} x^{3}\left(6656\theta^4+7680\theta^3-36960\theta^2-49920\theta-16985\right)-2^{15} x^{4}\left(512\theta^4+4864\theta^3+9136\theta^2+6464\theta+1587\right)+2^{20} x^{5}(4\theta+5)^2(4\theta+3)^2$

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Coefficients of the holomorphic solution: 1, -20, 684, -28496, 1317100, ...
--> OEIS
Normalized instanton numbers (n0=1): -44/5, -277/5, -596, -7236, -502128/5, ... ; Common denominator:...

#### Discriminant

$(1+16z)(64z+1)^2(64z-5)^2$

#### Local exponents

$-\frac{ 1}{ 16}$$-\frac{ 1}{ 64}$$0$$\frac{ 5}{ 64}$$\infty$
$0$$0$$0$$0$$\frac{ 3}{ 4}$
$1$$\frac{ 1}{ 2}$$0$$1$$\frac{ 3}{ 4}$
$1$$\frac{ 1}{ 2}$$0$$3$$\frac{ 5}{ 4}$
$2$$1$$0$$4$$\frac{ 5}{ 4}$

#### Note:

This is operator "5.50" from ...

3

New Number: 5.52 |  AESZ: 252  |  Superseeker: -232/5 -122168/5  |  Hash: cae57e93a6afb98313f62899d1f75e2e

Degree: 5

$5^{2} \theta^4-2^{2} 5 x\left(36\theta^4-636\theta^3-488\theta^2-170\theta-25\right)-2^{4} x^{2}\left(21301\theta^4+27148\theta^3-86889\theta^2-63110\theta-14975\right)+2^{8} 5 x^{3}\left(3907\theta^4-58863\theta^3-25285\theta^2+10878\theta+7151\right)+2^{10} 59 x^{4}\left(10981\theta^4-29878\theta^3-89811\theta^2-70372\theta-17759\right)+2^{15} 3 59^{2} x^{5}(4\theta+5)(3\theta+2)(3\theta+4)(4\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -20, 684, -32240, 1969900, ...
--> OEIS
Normalized instanton numbers (n0=1): -232/5, -7499/10, -122168/5, -4503443/5, -200467616/5, ... ; Common denominator:...

#### Discriminant

$(108z+1)(2048z^2+52z+1)(-5+472z)^2$

#### Local exponents

$-\frac{ 13}{ 1024}-\frac{ 7}{ 1024}\sqrt{ 7}I$$-\frac{ 13}{ 1024}+\frac{ 7}{ 1024}\sqrt{ 7}I$$-\frac{ 1}{ 108}$$0$$\frac{ 5}{ 472}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 2}{ 3}$
$1$$1$$1$$0$$1$$\frac{ 3}{ 4}$
$1$$1$$1$$0$$3$$\frac{ 5}{ 4}$
$2$$2$$2$$0$$4$$\frac{ 4}{ 3}$

#### Note:

This is operator "5.52" from ...

4

New Number: 5.64 |  AESZ: 272  |  Superseeker: 468/5 11885484  |  Hash: 467bb784f4bd6e978748e98f6ea4a573

Degree: 5

$5^{2} \theta^4-2^{2} 3 5 x\left(1332\theta^4+3528\theta^3+3289\theta^2+1525\theta+285\right)+2^{4} 3^{2} x^{2}\left(331776\theta^4+2602368\theta^3+4533336\theta^2+2996640\theta+724415\right)+2^{8} 3^{5} x^{3}\left(539136\theta^4+622080\theta^3-3024864\theta^2-4008960\theta-1315985\right)-2^{15} 3^{8} x^{4}\left(41472\theta^4+393984\theta^3+735984\theta^2+510912\theta+120811\right)-2^{20} 3^{11} x^{5}(12\theta+7)(12\theta+11)(12\theta+13)(12\theta+17)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 684, 761004, 985011120, 1373164693740, ...
--> OEIS
Normalized instanton numbers (n0=1): 468/5, -315477/5, 11885484, -14354122356/5, 808514230608, ... ; Common denominator:...

#### Discriminant

$-(-1+432z)(1728z+5)^2(1728z-1)^2$

#### Local exponents

$-\frac{ 5}{ 1728}$$0$$\frac{ 1}{ 1728}$$\frac{ 1}{ 432}$$\infty$
$0$$0$$0$$0$$\frac{ 7}{ 12}$
$1$$0$$-\frac{ 1}{ 6}$$1$$\frac{ 11}{ 12}$
$3$$0$$1$$1$$\frac{ 13}{ 12}$
$4$$0$$\frac{ 7}{ 6}$$2$$\frac{ 17}{ 12}$

#### Note:

This is operator "5.64" from ...