Calabi-Yau differential operator database v.3.0

1

New Number: 2.54 | AESZ: 41 | Superseeker: 2 -104 | Hash: a9ddeed4299f59fb9ac9f6f248383b8f

Degree: 2

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

Coefficients of the holomorphic solution: 1, 6, 54, 60, -19530, ...
--> OEIS
Normalized instanton numbers (n₀=1): 2, -7, -104, -588, 3300, ... ; Common denominator:...

Discriminant

$1-56z+1296z^2$

Local exponents

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

\(0\)	\(\frac{ 7}{ 324}-\frac{ 1}{ 81}\sqrt{ 2}I\)	\(\frac{ 7}{ 324}+\frac{ 1}{ 81}\sqrt{ 2}I\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $I \ast \delta$

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

2

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 $I \ast \iota$

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

3

New Number: 5.31 | AESZ: 212 | Superseeker: -20/7 -104 | Hash: f72aa947ba945355102b3fef56e0af0f

Degree: 5

$7^{2} \theta^4+2 7 x\left(134\theta^4+286\theta^3+234\theta^2+91\theta+14\right)-2^{2} x^{2}\left(3183\theta^4+10266\theta^3+13501\theta^2+8225\theta+1918\right)+2^{3} x^{3}\left(2588\theta^4+8400\theta^3+10256\theta^2+5649\theta+1190\right)-2^{4} 3 x^{4}\left(256\theta^4+848\theta^3+1141\theta^2+717\theta+174\right)+2^{8} 3^{2} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, -4, 64, -1408, 37216, ...
--> OEIS
Normalized instanton numbers (n₀=1): -20/7, 57/4, -104, 16385/14, -110508/7, ... ; Common denominator:...

Discriminant

$(4z-1)(16z^2-44z-1)(6z-7)^2$

Local exponents

$\frac{ 11}{ 8}-\frac{ 5}{ 8}\sqrt{ 5}$ $0$ $\frac{ 1}{ 4}$ $\frac{ 7}{ 6}$ $\frac{ 11}{ 8}+\frac{ 5}{ 8}\sqrt{ 5}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $1$
$1$ $0$ $1$ $1$ $1$ $1$
$1$ $0$ $1$ $3$ $1$ $1$
$2$ $0$ $2$ $4$ $2$ $1$

\(\frac{ 11}{ 8}-\frac{ 5}{ 8}\sqrt{ 5}\)	\(0\)	\(\frac{ 1}{ 4}\)	\(\frac{ 7}{ 6}\)	\(\frac{ 11}{ 8}+\frac{ 5}{ 8}\sqrt{ 5}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(0\)	\(1\)	\(3\)	\(1\)	\(1\)
\(2\)	\(0\)	\(2\)	\(4\)	\(2\)	\(1\)

Note:

There is a second MUM-point corresponding to Operator AESZ 117 /5.515.

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

4

New Number: 8.3 | AESZ: 105 | Superseeker: 8 -104 | Hash: 7b27135451cf2016217211c633b7ab83

Degree: 8

$\theta^4-2^{2} x(3\theta^2+3\theta+1)(7\theta^2+7\theta+2)+2^{5} 3 x^{2}\left(15\theta^4+28\theta^3+39\theta^2+22\theta+4\right)-2^{10} x^{3}\left(21\theta^4-126\theta^3-386\theta^2-291\theta-76\right)+2^{14} x^{4}\left(37\theta^4+74\theta^3+50\theta^2+13\theta+6\right)+2^{18} x^{5}\left(21\theta^4+210\theta^3+118\theta^2-19\theta-24\right)+2^{21} 3 x^{6}\left(15\theta^4+32\theta^3+45\theta^2+32\theta+8\right)+2^{26} x^{7}(3\theta^2+3\theta+1)(7\theta^2+7\theta+2)+2^{32} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 200, 6272, 233896, ...
--> OEIS
Normalized instanton numbers (n₀=1): 8, 71/2, -104, 4202, 50112, ... ; Common denominator:...

Discriminant

$(8z+1)(64z-1)(4z+1)(32z-1)(1+256z^2)^2$

Local exponents

$-\frac{ 1}{ 4}$ $-\frac{ 1}{ 8}$ $0-\frac{ 1}{ 16}I$ $0$ $0+\frac{ 1}{ 16}I$ $\frac{ 1}{ 64}$ $\frac{ 1}{ 32}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$
$1$ $1$ $3$ $0$ $3$ $1$ $1$ $1$
$2$ $2$ $4$ $0$ $4$ $2$ $2$ $1$

\(-\frac{ 1}{ 4}\)	\(-\frac{ 1}{ 8}\)	\(0-\frac{ 1}{ 16}I\)	\(0\)	\(0+\frac{ 1}{ 16}I\)	\(\frac{ 1}{ 64}\)	\(\frac{ 1}{ 32}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(3\)	\(0\)	\(3\)	\(1\)	\(1\)	\(1\)
\(2\)	\(2\)	\(4\)	\(0\)	\(4\)	\(2\)	\(2\)	\(1\)

Note:

Hadamard product $a \ast d$. 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
$Q(\sqrt{-1})$.

\(.872284041I\)	\(-\frac{ 1}{ 6}+30\lambda\)	\(-\frac{ 1}{ 12}+15\lambda\)	\(-\frac{ 1}{ 243}\sqrt{ 3}ln(2)^2/(-8I\lambdaPi^3)^(\frac{ 1}{ 6})+5\lambda\)
\(6\)	\(2\)	\(\frac{ 1}{ 2}\)	\(\frac{ 1}{ 12}-15\lambda\)
\(-12\)	\(-2\)	\(0\)	\(-\frac{ 1}{ 6}+30\lambda\)
\(72\)	\(12\)	\(6\)	\(2.-.872284041I\)

\(2.+.872284041I\)	\(-\frac{ 1}{ 6}-30\lambda\)	\(\frac{ 1}{ 12}+15\lambda\)	\(-\frac{ 1}{ 243}\sqrt{ 3}ln(2)^2/(-8I\lambdaPi^3)^(\frac{ 1}{ 6})-5\lambda\)
\(6\)	\(0\)	\(\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 12}-15\lambda\)
\(12\)	\(-2\)	\(2\)	\(-\frac{ 1}{ 6}-30\lambda\)
\(72\)	\(-12\)	\(6\)	\(-.872284041I\)

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Note:

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:

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

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

\(1-\frac{ 872284041}{ 1000000000}I\)	\(24\)	\(\frac{ 7}{ 6}\)	\(1\)
\(-6\)	\(-48\)	\(-1\)	\(0\)
\(12\)	\(72\)	\(0\)	\(0\)
\(-72\)	\(0\)	\(0\)	\(0\)

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

\(\frac{ 1}{ 2}-36\lambda\)	\(-3\)	\(\frac{ 5}{ 6}\)	\(1\)
\(0\)	\(12\)	\(-1\)	\(0\)
\(6\)	\(-36\)	\(0\)	\(0\)
\(36\)	\(0\)	\(0\)	\(0\)