Calabi-Yau differential operator database v.3.0

1

New Number: 2.13 | AESZ: 36 | Superseeker: 16 1232 | Hash: dea6fdf568a5907a24ba30fef2caf124

Degree: 2

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

Coefficients of the holomorphic solution: 1, 16, 720, 44800, 3312400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 16, 42, 1232, 32159, 990128, ... ; Common denominator:...

Discriminant

$(128z-1)(64z-1)$

Local exponents

$0$ $\frac{ 1}{ 128}$ $\frac{ 1}{ 64}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $\frac{ 3}{ 2}$
$0$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 128}\)	\(\frac{ 1}{ 64}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

A*d

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-88\)	\(80\)	\(32\)	\(12\)

2

New Number: 2.55 | AESZ: 42 | Superseeker: 8 1000 | Hash: c389d3bc0e31801bc4b7b3e186702bc9

Degree: 2

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 240, 10880, 597520, ...
--> OEIS
Normalized instanton numbers (n₀=1): 8, 63, 1000, 44369/2, 606168, ... ; Common denominator:...

Discriminant

$1-96z+256z^2$

Local exponents

$0$ $\frac{ 3}{ 16}-\frac{ 1}{ 8}\sqrt{ 2}$ $\frac{ 3}{ 16}+\frac{ 1}{ 8}\sqrt{ 2}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$
$0$ $1$ $1$ $1$
$0$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 3}{ 16}-\frac{ 1}{ 8}\sqrt{ 2}\)	\(\frac{ 3}{ 16}+\frac{ 1}{ 8}\sqrt{ 2}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $I \ast \epsilon$

	Gopakumar-Vafa invariants
g=0	256, 2016, 32000, 709904, 19397376, 604985440, 20693162240, 757512952944,...
g=1	0, 0, 0, -8, 1024, 1220032, 162833408, 14777331408,...
g=2	,...

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-116\)	\(80\)	\(32\)	\(12\)

3

New Number: 5.11 | AESZ: 71 | Superseeker: 112 378800 | Hash: cf4de65b0566a4f6294132c167d227eb

Degree: 5

$\theta^4+2^{4} x\left(39\theta^4-42\theta^3-29\theta^2-8\theta-1\right)+2^{11} x^{2}\theta(37\theta^3-137\theta^2-10\theta-1)-2^{16} x^{3}\left(181\theta^4+456\theta^3+353\theta^2+132\theta+19\right)-2^{23} 5 x^{4}\left(36\theta^4+60\theta^3+36\theta^2+6\theta-1\right)+2^{30} 5^{2} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 16, 656, 40192, 3006736, ...
--> OEIS
Normalized instanton numbers (n₀=1): 112, -4570, 378800, -40565898, 5098744272, ... ; Common denominator:...

Discriminant

$(16z-1)(128z-1)(128z+1)(1+320z)^2$

Local exponents

$-\frac{ 1}{ 128}$ $-\frac{ 1}{ 320}$ $0$ $\frac{ 1}{ 128}$ $\frac{ 1}{ 16}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $0$ $1$ $1$ $1$
$1$ $3$ $0$ $1$ $1$ $1$
$2$ $4$ $0$ $2$ $2$ $1$

\(-\frac{ 1}{ 128}\)	\(-\frac{ 1}{ 320}\)	\(0\)	\(\frac{ 1}{ 128}\)	\(\frac{ 1}{ 16}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(1\)	\(3\)	\(0\)	\(1\)	\(1\)	\(1\)
\(2\)	\(4\)	\(0\)	\(2\)	\(2\)	\(1\)

Note:

This is operator "5.11" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(304\)	\(32\)	\(32\)	\(8\)

4

New Number: 5.13 | AESZ: 83 | Superseeker: -80 -174096 | Hash: 171e1251d8e4f7de878d0d07de6f58ab

Degree: 5

$\theta^4-2^{4} x\left(88\theta^4+32\theta^3+33\theta^2+17\theta+3\right)+2^{9} x^{2}\left(1504\theta^4+1408\theta^3+1436\theta^2+596\theta+93\right)-2^{18} x^{3}\left(776\theta^4+1344\theta^3+1381\theta^2+651\theta+117\right)+2^{23} 3 x^{4}(2\theta+1)(512\theta^3+1152\theta^2+1054\theta+339)-2^{31} 3^{2} x^{5}(2\theta+1)(4\theta+3)(4\theta+5)(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 48, 5328, 779520, 131619600, ...
--> OEIS
Normalized instanton numbers (n₀=1): -80, -2954, -174096, -13270953, -1179175536, ... ; Common denominator:...

Discriminant

$-(128z-1)(384z-1)^2(256z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 384}$ $\frac{ 1}{ 256}$ $\frac{ 1}{ 128}$ $\infty$
$0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $\frac{ 1}{ 2}$ $1$ $\frac{ 3}{ 4}$
$0$ $3$ $\frac{ 1}{ 2}$ $1$ $\frac{ 5}{ 4}$
$0$ $4$ $1$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 384}\)	\(\frac{ 1}{ 256}\)	\(\frac{ 1}{ 128}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(\frac{ 1}{ 2}\)	\(1\)	\(\frac{ 3}{ 4}\)
\(0\)	\(3\)	\(\frac{ 1}{ 2}\)	\(1\)	\(\frac{ 5}{ 4}\)
\(0\)	\(4\)	\(1\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.13" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(360\)	\(-16\)	\(32\)	\(4\)

5

New Number: 8.19 | AESZ: 201 | Superseeker: 32 7584 | Hash: d21570c07bca6887061716b2d727fa75

Degree: 8

$\theta^4-2^{4} x\left(13\theta^4+26\theta^3+20\theta^2+7\theta+1\right)+2^{8} x^{2}\left(9\theta^4+192\theta^3+249\theta^2+114\theta+20\right)+2^{12} x^{3}\left(379\theta^4+246\theta^3-569\theta^2-318\theta-60\right)-2^{16} x^{4}\left(749\theta^4+2560\theta^3-1722\theta^2-1862\theta-474\right)-2^{20} 13 x^{5}\left(251\theta^4-10\theta^3+262\theta^2+145\theta+27\right)+2^{24} 13 x^{6}\left(471\theta^4+96\theta^3+17\theta^2+96\theta+42\right)+2^{28} 13^{2} x^{7}\left(41\theta^4+82\theta^3+67\theta^2+26\theta+4\right)-2^{35} 13^{2} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 16, 752, 49408, 3805456, ...
--> OEIS
Normalized instanton numbers (n₀=1): 32, -152, 7584, -160593, 7055200, ... ; Common denominator:...

Discriminant

$-(128z-1)(16z+1)(256z^2-96z+1)(-1+3328z^2)^2$

Local exponents

$-\frac{ 1}{ 16}$ $-\frac{ 1}{ 208}\sqrt{ 13}$ $0$ $\frac{ 1}{ 128}$ $\frac{ 3}{ 16}-\frac{ 1}{ 8}\sqrt{ 2}$ $\frac{ 1}{ 208}\sqrt{ 13}$ $\frac{ 3}{ 16}+\frac{ 1}{ 8}\sqrt{ 2}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $0$ $1$ $1$ $1$ $1$ $1$
$1$ $3$ $0$ $1$ $1$ $3$ $1$ $1$
$2$ $4$ $0$ $2$ $2$ $4$ $2$ $1$

\(-\frac{ 1}{ 16}\)	\(-\frac{ 1}{ 208}\sqrt{ 13}\)	\(0\)	\(\frac{ 1}{ 128}\)	\(\frac{ 3}{ 16}-\frac{ 1}{ 8}\sqrt{ 2}\)	\(\frac{ 1}{ 208}\sqrt{ 13}\)	\(\frac{ 3}{ 16}+\frac{ 1}{ 8}\sqrt{ 2}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(3\)	\(0\)	\(1\)	\(1\)	\(3\)	\(1\)	\(1\)
\(2\)	\(4\)	\(0\)	\(2\)	\(2\)	\(4\)	\(2\)	\(1\)

Note:

This operator has a second MUM-point, corresponding to operator 8.18

\(\frac{ 10}{ 3}+176\lambda\)	\(-\frac{ 7}{ 12}-44\lambda\)	\(\frac{ 35}{ 144}+\frac{ 55}{ 3}\lambda\)	\(-.73703221e-1-\frac{ 77}{ 6}\lambda\)
\(\frac{ 20}{ 3}+\frac{ 1}{ 500000000}I\)	\(-\frac{ 2}{ 3}+\frac{ 1}{ 1000000000}I\)	\(\frac{ 25}{ 36}\)	\(-\frac{ 35}{ 144}-\frac{ 55}{ 3}\lambda\)
\(16+\frac{ 1}{ 200000000}I\)	\(-4+\frac{ 1}{ 500000000}I\)	\(\frac{ 8}{ 3}-\frac{ 1}{ 500000000}I\)	\(-\frac{ 7}{ 12}-44\lambda\)
\(64.+.23e-7I\)	\(-15.999999981+.90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-8I\)	\(\frac{ 20}{ 3}-\frac{ 1}{ 125000000}I\)	\(-\frac{ 4}{ 3}-176\lambda\)

\(9.+5.059247438I\)	\(-\frac{ 8}{ 3}-348\lambda\)	\(.83333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333+.527004941I\)	\(-.133347276-58\lambda\)
\(30\)	\(-9\)	\(\frac{ 25}{ 8}\)	\(-.83333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333-.527004941I\)
\(96\)	\(-32\)	\(11\)	\(-\frac{ 8}{ 3}-348\lambda\)
\(288+\frac{ 1}{ 1000000000}I\)	\(-96\)	\(30\)	\(-7.-5.059247438I\)

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Note:

Explicit solution

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

\(1\)	\(-1\)	\(\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 6}\)
\(0\)	\(1\)	\(-1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(0\)	\(1\)	\(-1\)
\(0\)	\(0\)	\(0\)	\(1\)

\(1+88\lambda\)	\(0\)	\(\frac{ 55}{ 6}\lambda\)	\(.5683112e-2\)
\(\frac{ 10}{ 3}\)	\(1\)	\(\frac{ 25}{ 72}\)	\(-\frac{ 55}{ 6}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(32\)	\(0\)	\(\frac{ 10}{ 3}\)	\(1-88\lambda\)

\(-88\lambda\)	\(\frac{ 26}{ 3}\)	\(1\)	\(1\)
\(-\frac{ 10}{ 3}\)	\(-16\)	\(-1\)	\(0\)
\(0\)	\(32\)	\(0\)	\(0\)
\(-32\)	\(0\)	\(0\)	\(0\)

\(1+116\lambda\)	\(0\)	\(\frac{ 1}{ 343}I5^{ \frac{ 7}{ 9}}7^{ \frac{ 1}{ 3}}ln(2)^3\)	\(.9874994e-2\)
\(\frac{ 10}{ 3}\)	\(1\)	\(\frac{ 25}{ 72}\)	\(-\frac{ 1}{ 343}I5^{ \frac{ 7}{ 9}}7^{ \frac{ 1}{ 3}}ln(2)^3\)
\(0\)	\(0\)	\(1\)	\(0\)
\(32\)	\(0\)	\(\frac{ 10}{ 3}\)	\(1-116\lambda\)

\(-116\lambda\)	\(\frac{ 26}{ 3}\)	\(1\)	\(1\)
\(-\frac{ 10}{ 3}\)	\(-16\)	\(-1\)	\(0\)
\(0\)	\(32\)	\(0\)	\(0\)
\(-32\)	\(0\)	\(0\)	\(0\)