Calabi-Yau differential operator database v.3.0

1

New Number: 2.18 | AESZ: 110 | Superseeker: 36 8076 | Hash: 5060b638cac581d5f0f9dd7f40d90e6c

Degree: 2

$\theta^4-2^{2} 3 x(3\theta+1)(3\theta+2)(8\theta^2+8\theta+3)+2^{8} 3^{2} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 72, 14760, 3951360, 1198751400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 36, -144, 8076, -57996, 6960672, ... ; Common denominator:...

Discriminant

$(432z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 432}$ $\infty$
$0$ $0$ $\frac{ 1}{ 3}$
$0$ $\frac{ 1}{ 2}$ $\frac{ 2}{ 3}$
$0$ $\frac{ 1}{ 2}$ $\frac{ 4}{ 3}$
$0$ $1$ $\frac{ 5}{ 3}$

\(0\)	\(\frac{ 1}{ 432}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(0\)	\(\frac{ 1}{ 2}\)	\(\frac{ 2}{ 3}\)
\(0\)	\(\frac{ 1}{ 2}\)	\(\frac{ 4}{ 3}\)
\(0\)	\(1\)	\(\frac{ 5}{ 3}\)

2

New Number: 2.22 | AESZ: 135 | Superseeker: 36 -206716/3 | Hash: 85e55291bd94bb32087b43f104c60645

Degree: 2

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 36, 3780, 388080, 8108100, ...
--> OEIS
Normalized instanton numbers (n₀=1): 36, -477, -206716/3, -4431924, -27005472, ... ; Common denominator:...

Discriminant

$1-576z+110592z^2$

Local exponents

$0$ $\frac{ 1}{ 384}-\frac{ 1}{ 1152}\sqrt{ 3}I$ $\frac{ 1}{ 384}+\frac{ 1}{ 1152}\sqrt{ 3}I$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 4}$
$0$ $1$ $1$ $\frac{ 3}{ 4}$
$0$ $1$ $1$ $\frac{ 5}{ 4}$
$0$ $2$ $2$ $\frac{ 7}{ 4}$

\(0\)	\(\frac{ 1}{ 384}-\frac{ 1}{ 1152}\sqrt{ 3}I\)	\(\frac{ 1}{ 384}+\frac{ 1}{ 1152}\sqrt{ 3}I\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 3}{ 4}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 5}{ 4}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 7}{ 4}\)

Integral instantons: Hadamard, product, $C, \ast, f$.,...
Coefficients of the Yukawa coupling: 1, 36, -3780, -1860408, -283646916, -3375683964, 8182006866936, 1873126855818072,...
Coefficients of the q-coordinate : 0, 1, -156, 22446, -2570800, 297228021, -32847734088, 3343576302482,...

	Gopakumar-Vafa invariants
g=0	,...
g=1	,...
g=2	,...

No topological data

Monodromy (with respect to Frobenius basis)

$1$	$-1$	$\frac{ 1}{ 2}$	$-\frac{ 1}{ 6}$
$0$	$1$	$-1$	$\frac{ 1}{ 2}$
$0$	$0$	$1$	$-1$
$0$	$0$	$0$	$1$

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$\frac{ 5}{ 24}+Itan(\frac{ 801}{ 812})$	$-\frac{ 19}{ 144}+52\lambda$	$-\frac{ 323}{ 1728}+\frac{ 221}{ 3}\lambda$	$.92182440e-1+\frac{ 247}{ 9}\lambda$
$\frac{ 17}{ 4}+\frac{ 9}{ 1000000000}I$	$\frac{ 41}{ 24}-\frac{ 3}{ 1000000000}I$	$\frac{ 289}{ 288}-\frac{ 1}{ 250000000}I$	$\frac{ 323}{ 1728}-\frac{ 221}{ 3}\lambda$
$-3-\frac{ 3}{ 500000000}I$	$-\frac{ 1}{ 2}+\frac{ 1}{ 500000000}I$	$\frac{ 7}{ 24}+\frac{ 3}{ 1000000000}I$	$-\frac{ 19}{ 144}+52\lambda$
$18.000000018+.41e-7I$	$3.000000039-.12000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-7I$	$\frac{ 17}{ 4}-\frac{ 9}{ 500000000}I$	$\frac{ 43}{ 24}-Itan(\frac{ 801}{ 812})$

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$\frac{ 43}{ 24}+Itan(\frac{ 801}{ 812})$	$-\frac{ 19}{ 144}-52\lambda$	$\frac{ 323}{ 1728}+\frac{ 221}{ 3}\lambda$	$.92182440e-1-\frac{ 247}{ 9}\lambda$
$\frac{ 17}{ 4}$	$\frac{ 7}{ 24}$	$\frac{ 289}{ 288}$	$-\frac{ 323}{ 1728}-\frac{ 221}{ 3}\lambda$
$3$	$-\frac{ 1}{ 2}$	$\frac{ 41}{ 24}$	$-\frac{ 19}{ 144}-52\lambda$
$18$	$-3$	$\frac{ 17}{ 4}$	$\frac{ 5}{ 24}-Itan(\frac{ 801}{ 812})$

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Basis of the Doran-Morgan lattice

$\frac{ 19}{ 24}-\frac{ 1162393719353719131361189363784404603456366390101814206736549803033200099854000208071729302850101578}{ 768799759549186043811713254203741173211443985818991590275710787225120378790852066470688698366857287}I$	$\frac{ 8750000003}{ 1000000000}+\frac{ 113}{ 6000000000}I$	$\frac{ 378000000701999998397}{ 324000000648000002005}+\frac{ 1707000001584}{ 324000000648000002005}I$	$1$
$-\frac{ 17}{ 4}-\frac{ 9}{ 1000000000}I$	$-\frac{ 12000000009}{ 1000000000}-\frac{ 53}{ 2000000000}I$	$-\frac{ 324000000647999998643}{ 324000000648000002005}-\frac{ 1476000001476}{ 324000000648000002005}I$	$0$
$3+\frac{ 3}{ 500000000}I$	$\frac{ 9000000009}{ 500000000}+\frac{ 41}{ 1000000000}I$	$0$	$0$
$-\frac{ 9000000009}{ 500000000}-\frac{ 41}{ 1000000000}I$	$0$	$0$	$0$

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3

New Number: 2.58 | AESZ: 46 | Superseeker: -6 -104 | Hash: 2226ec115674e71c483ba2c0350e8adf

Degree: 2

$\theta^4-2 3 x(2\theta+1)^2(9\theta^2+9\theta+5)+2^{2} 3^{6} x^{2}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 30, 1782, 129900, 10463670, ...
--> OEIS
Normalized instanton numbers (n₀=1): -6, -6, -104, 36, -4812, ... ; Common denominator:...

Discriminant

$(108z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 108}$ $\infty$
$0$ $0$ $\frac{ 1}{ 2}$
$0$ $\frac{ 1}{ 6}$ $1$
$0$ $\frac{ 5}{ 6}$ $1$
$0$ $1$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 108}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(\frac{ 1}{ 6}\)	\(1\)
\(0\)	\(\frac{ 5}{ 6}\)	\(1\)
\(0\)	\(1\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $I \ast \iota$

4

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

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

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

Note:

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

	Gopakumar-Vafa invariants
g=0	540, 12555, 621315, 44892765, 3995437590, 406684089360, 45426958360155, 5432556927598425,...
g=1	0, 0, -1, 13095, 17230617, 6648808835, 1831575868830, 433375127634753,...
g=2	,...

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-150\)	\(66\)	\(15\)	\(8\)

5

New Number: 3.29 | AESZ: 411 | Superseeker: 3 237 | Hash: 767c4e8d5a7bc53fbbd0d49797e65358

Degree: 3

$\theta^4-x\left(16+98\theta+235\theta^2+274\theta^3+145\theta^4\right)+2^{3} x^{2}(2\theta+1)(4\theta+5)(97\theta^2+190\theta+120)-2^{4} 3^{4} x^{3}(4\theta+5)(2\theta+3)(2\theta+1)(4\theta+9)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 16, 468, 17520, 774060, ...
--> OEIS
Normalized instanton numbers (n₀=1): 3, 36, 237, 4638, 72330, ... ; Common denominator:...

Discriminant

$-(81z-1)(-1+32z)^2$

Local exponents

$0$ $\frac{ 1}{ 81}$ $\frac{ 1}{ 32}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $\frac{ 1}{ 4}$ $\frac{ 5}{ 4}$
$0$ $1$ $1$ $\frac{ 3}{ 2}$
$0$ $2$ $\frac{ 5}{ 4}$ $\frac{ 9}{ 4}$

\(0\)	\(\frac{ 1}{ 81}\)	\(\frac{ 1}{ 32}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(\frac{ 1}{ 4}\)	\(\frac{ 5}{ 4}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(0\)	\(2\)	\(\frac{ 5}{ 4}\)	\(\frac{ 9}{ 4}\)

Note:

This is operator "3.29" from ...

\(-\frac{ 1}{ 2}+24\lambda\)	\(12\lambda\)	\(-\frac{ 1}{ 8}+2\lambda\)	\(-.1127229e-2\)
\(-1\)	\(\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 12}\)	\(-\frac{ 1}{ 8}-2\lambda\)
\(-6\)	\(0\)	\(\frac{ 1}{ 2}\)	\(-12\lambda\)
\(-12\)	\(-6\)	\(-1\)	\(-\frac{ 1}{ 2}-24\lambda\)

\(\frac{ 5}{ 24}+Itan(\frac{ 801}{ 812})\)	\(-\frac{ 19}{ 144}+52\lambda\)	\(-\frac{ 323}{ 1728}+\frac{ 221}{ 3}\lambda\)	\(.92182440e-1+\frac{ 247}{ 9}\lambda\)
\(\frac{ 17}{ 4}+\frac{ 9}{ 1000000000}I\)	\(\frac{ 41}{ 24}-\frac{ 3}{ 1000000000}I\)	\(\frac{ 289}{ 288}-\frac{ 1}{ 250000000}I\)	\(\frac{ 323}{ 1728}-\frac{ 221}{ 3}\lambda\)
\(-3-\frac{ 3}{ 500000000}I\)	\(-\frac{ 1}{ 2}+\frac{ 1}{ 500000000}I\)	\(\frac{ 7}{ 24}+\frac{ 3}{ 1000000000}I\)	\(-\frac{ 19}{ 144}+52\lambda\)
\(18.000000018+.41e-7I\)	\(3.000000039-.12000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-7I\)	\(\frac{ 17}{ 4}-\frac{ 9}{ 500000000}I\)	\(\frac{ 43}{ 24}-Itan(\frac{ 801}{ 812})\)

\(\frac{ 43}{ 24}+Itan(\frac{ 801}{ 812})\)	\(-\frac{ 19}{ 144}-52\lambda\)	\(\frac{ 323}{ 1728}+\frac{ 221}{ 3}\lambda\)	\(.92182440e-1-\frac{ 247}{ 9}\lambda\)
\(\frac{ 17}{ 4}\)	\(\frac{ 7}{ 24}\)	\(\frac{ 289}{ 288}\)	\(-\frac{ 323}{ 1728}-\frac{ 221}{ 3}\lambda\)
\(3\)	\(-\frac{ 1}{ 2}\)	\(\frac{ 41}{ 24}\)	\(-\frac{ 19}{ 144}-52\lambda\)
\(18\)	\(-3\)	\(\frac{ 17}{ 4}\)	\(\frac{ 5}{ 24}-Itan(\frac{ 801}{ 812})\)

\(\frac{ 19}{ 24}-\frac{ 1162393719353719131361189363784404603456366390101814206736549803033200099854000208071729302850101578}{ 768799759549186043811713254203741173211443985818991590275710787225120378790852066470688698366857287}I\)	\(\frac{ 8750000003}{ 1000000000}+\frac{ 113}{ 6000000000}I\)	\(\frac{ 378000000701999998397}{ 324000000648000002005}+\frac{ 1707000001584}{ 324000000648000002005}I\)	\(1\)
\(-\frac{ 17}{ 4}-\frac{ 9}{ 1000000000}I\)	\(-\frac{ 12000000009}{ 1000000000}-\frac{ 53}{ 2000000000}I\)	\(-\frac{ 324000000647999998643}{ 324000000648000002005}-\frac{ 1476000001476}{ 324000000648000002005}I\)	\(0\)
\(3+\frac{ 3}{ 500000000}I\)	\(\frac{ 9000000009}{ 500000000}+\frac{ 41}{ 1000000000}I\)	\(0\)	\(0\)
\(-\frac{ 9000000009}{ 500000000}-\frac{ 41}{ 1000000000}I\)	\(0\)	\(0\)	\(0\)

\(\frac{ 1}{ 2}+36\lambda\)	\(\frac{ 1}{ 4}+6\lambda\)	\(0\)	\(.19987912e-1\)
\(0\)	\(1\)	\(0\)	\(0\)
\(-6\)	\(3\)	\(1\)	\(\frac{ 1}{ 4}-6\lambda\)
\(-36\)	\(-6\)	\(0\)	\(\frac{ 1}{ 2}-36\lambda\)

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Explicit solution

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Explicit solution

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

\(\frac{ 3}{ 2}-24\lambda\)	\(0\)	\(\frac{ 1}{ 2}\)	\(1\)
\(1\)	\(0\)	\(-1\)	\(0\)
\(6\)	\(-12\)	\(0\)	\(0\)
\(12\)	\(0\)	\(0\)	\(0\)

\(\frac{ 11}{ 2}+600\lambda\)	\(-\frac{ 9}{ 4}-300\lambda\)	\(\frac{ 21}{ 20}+140\lambda\)	\(-.196596398-90\lambda\)
\(14\)	\(-6\)	\(\frac{ 49}{ 15}\)	\(-\frac{ 21}{ 20}-140\lambda\)
\(30\)	\(-15\)	\(8\)	\(-\frac{ 9}{ 4}-300\lambda\)
\(60\)	\(-30\)	\(14\)	\(-\frac{ 7}{ 2}-600\lambda\)

\(1+150\lambda\)	\(0\)	\(\frac{ 55}{ 2}\lambda\)	\(.35225900e-1\)
\(\frac{ 11}{ 4}\)	\(1\)	\(\frac{ 121}{ 240}\)	\(-\frac{ 55}{ 2}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(15\)	\(0\)	\(\frac{ 11}{ 4}\)	\(1-150\lambda\)

\(-150\lambda\)	\(\frac{ 21}{ 4}\)	\(1\)	\(1\)
\(-\frac{ 11}{ 4}\)	\(-\frac{ 15}{ 2}\)	\(-1\)	\(0\)
\(0\)	\(15\)	\(0\)	\(0\)
\(-15\)	\(0\)	\(0\)	\(0\)