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: 2.19 | AESZ: 112 | Superseeker: -288 -96055968 | Hash: 9a988f0cb0ca922885043cdadf98dd79

Degree: 2

\(\theta^4-2^{4} 3 x(6\theta+1)(6\theta+5)(8\theta^2+8\theta+3)+2^{12} 3^{2} x^{2}(6\theta+1)(6\theta+5)(6\theta+7)(6\theta+11)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 720, 2273040, 9605756160, 46308725583120, ...
--> OEIS
Normalized instanton numbers (n₀=1): -288, 162504, -96055968, 106571782296, -135291308081760, ... ; Common denominator:...

Discriminant

\((6912z-1)^2\)

Local exponents

\(0\) \(\frac{ 1}{ 6912}\) \(\infty\)
\(0\) \(0\) \(\frac{ 1}{ 6}\)
\(0\) \(\frac{ 1}{ 2}\) \(\frac{ 5}{ 6}\)
\(0\) \(\frac{ 1}{ 2}\) \(\frac{ 7}{ 6}\)
\(0\) \(1\) \(\frac{ 11}{ 6}\)

\(0\)	\(\frac{ 1}{ 6912}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 6}\)
\(0\)	\(\frac{ 1}{ 2}\)	\(\frac{ 5}{ 6}\)
\(0\)	\(\frac{ 1}{ 2}\)	\(\frac{ 7}{ 6}\)
\(0\)	\(1\)	\(\frac{ 11}{ 6}\)

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

3

New Number: 3.15 | AESZ: | Superseeker: -32 -16288 | Hash: e21c92d8f9a2222be40fdc71ea51ee35

Degree: 3

\(\theta^4+2^{3} x\left(21\theta^4+42\theta^3+30\theta^2+9\theta+1\right)-2^{6} x^{2}(\theta+1)^2(96\theta^2+192\theta+77)+2^{9} 5^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, -8, 720, -68480, 8123920, ...
--> OEIS
Normalized instanton numbers (n₀=1): -32, 468, -16288, 1681645/2, -53608288, ... ; Common denominator:...

Discriminant

\((200z+1)(-1+16z)^2\)

Local exponents

\(-\frac{ 1}{ 200}\) \(0\) \(\frac{ 1}{ 16}\) \(\infty\)
\(0\) \(0\) \(0\) \(\frac{ 1}{ 2}\)
\(1\) \(0\) \(\frac{ 1}{ 6}\) \(1\)
\(1\) \(0\) \(\frac{ 5}{ 6}\) \(2\)
\(2\) \(0\) \(1\) \(\frac{ 5}{ 2}\)

\(-\frac{ 1}{ 200}\)	\(0\)	\(\frac{ 1}{ 16}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(\frac{ 1}{ 6}\)	\(1\)
\(1\)	\(0\)	\(\frac{ 5}{ 6}\)	\(2\)
\(2\)	\(0\)	\(1\)	\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to AESZ 328

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

4

New Number: 5.108 | AESZ: 365 | Superseeker: 4 1268 | Hash: f84624e83cd4eb2cc90693bd5627efcf

Degree: 5

\(\theta^4-2^{2} x\left(99\theta^4+78\theta^3+65\theta^2+26\theta+4\right)+2^{6} x^{2}\left(938\theta^4+1382\theta^3+1269\theta^2+554\theta+92\right)-2^{10} x^{3}\left(4171\theta^4+8736\theta^3+8690\theta^2+3948\theta+680\right)+2^{15} 5 x^{4}(2\theta+1)(418\theta^3+951\theta^2+846\theta+260)-2^{19} 3 5^{2} x^{5}(2\theta+1)(3\theta+2)(3\theta+4)(2\theta+3)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 16, 720, 44800, 3245200, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 107/2, 1268, 89439/4, 396908, ... ; Common denominator:...

Discriminant

\(-(108z-1)(2048z^2-128z+1)(-1+80z)^2\)

Local exponents

\(0\) \(\frac{ 1}{ 32}-\frac{ 1}{ 64}\sqrt{ 2}\) \(\frac{ 1}{ 108}\) \(\frac{ 1}{ 80}\) \(\frac{ 1}{ 32}+\frac{ 1}{ 64}\sqrt{ 2}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(\frac{ 1}{ 2}\)
\(0\) \(1\) \(1\) \(1\) \(1\) \(\frac{ 2}{ 3}\)
\(0\) \(1\) \(1\) \(3\) \(1\) \(\frac{ 4}{ 3}\)
\(0\) \(2\) \(2\) \(4\) \(2\) \(\frac{ 3}{ 2}\)

\(0\)	\(\frac{ 1}{ 32}-\frac{ 1}{ 64}\sqrt{ 2}\)	\(\frac{ 1}{ 108}\)	\(\frac{ 1}{ 80}\)	\(\frac{ 1}{ 32}+\frac{ 1}{ 64}\sqrt{ 2}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 2}{ 3}\)
\(0\)	\(1\)	\(1\)	\(3\)	\(1\)	\(\frac{ 4}{ 3}\)
\(0\)	\(2\)	\(2\)	\(4\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.108" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-36\)	\(112\)	\(64\)	\(20\)

5

New Number: 5.34 | AESZ: 217 | Superseeker: 17/7 5095/21 | Hash: e8743aeac19deca699ff90aaef6b8ea7

Degree: 5

\(7^{2} \theta^4+7 x\theta(-14-73\theta-118\theta^2+13\theta^3)-2^{3} 3 x^{2}\left(3378\theta^4+13446\theta^3+18869\theta^2+11158\theta+2352\right)-2^{4} 3^{3} x^{3}\left(3628\theta^4+17920\theta^3+31668\theta^2+22596\theta+5383\right)-2^{8} 3^{3} x^{4}(2\theta+1)(572\theta^3+2370\theta^2+2896\theta+1095)-2^{10} 3^{4} x^{5}(2\theta+1)(6\theta+5)(6\theta+7)(2\theta+3)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 72, 720, 37800, ...
--> OEIS
Normalized instanton numbers (n₀=1): 17/7, 254/7, 5095/21, 29600/7, 491991/7, ... ; Common denominator:...

Discriminant

\(-(16z+1)(27z+1)(48z-1)(7+24z)^2\)

Local exponents

\(-\frac{ 7}{ 24}\) \(-\frac{ 1}{ 16}\) \(-\frac{ 1}{ 27}\) \(0\) \(\frac{ 1}{ 48}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(\frac{ 1}{ 2}\)
\(1\) \(1\) \(1\) \(0\) \(1\) \(\frac{ 5}{ 6}\)
\(3\) \(1\) \(1\) \(0\) \(1\) \(\frac{ 7}{ 6}\)
\(4\) \(2\) \(2\) \(0\) \(2\) \(\frac{ 3}{ 2}\)

\(-\frac{ 7}{ 24}\)	\(-\frac{ 1}{ 16}\)	\(-\frac{ 1}{ 27}\)	\(0\)	\(\frac{ 1}{ 48}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(\frac{ 5}{ 6}\)
\(3\)	\(1\)	\(1\)	\(0\)	\(1\)	\(\frac{ 7}{ 6}\)
\(4\)	\(2\)	\(2\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.34" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-112\)	\(84\)	\(42\)	\(14\)

6

New Number: 1.12 | AESZ: 12 | Superseeker: 7776 66942277344 | Hash: ad7e2e881b3939396323eb746eb17a58

Degree: 1

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 720, 5821200, 75473798400, 1205906199498000, ...
--> OEIS
Normalized instanton numbers (n₀=1): 7776, 13952088, 66942277344, 475338414733416, 4184555647748620320, ... ; Common denominator:...

Discriminant

\(1-27648z\)

Local exponents

\(0\) \(\frac{ 1}{ 27648}\) \(\infty\)
\(0\) \(0\) \(\frac{ 1}{ 6}\)
\(0\) \(1\) \(\frac{ 1}{ 4}\)
\(0\) \(1\) \(\frac{ 3}{ 4}\)
\(0\) \(2\) \(\frac{ 5}{ 6}\)

\(0\)	\(\frac{ 1}{ 27648}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 6}\)
\(0\)	\(1\)	\(\frac{ 1}{ 4}\)
\(0\)	\(1\)	\(\frac{ 3}{ 4}\)
\(0\)	\(2\)	\(\frac{ 5}{ 6}\)

Note:

A-incarnation: X(3,4) in P^5(1,1,1,1,1,2)

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

\(-.16666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666667-.969204490I\)	\(-100\lambda\)	\(-\frac{ 35}{ 72}-\frac{ 250}{ 3}\lambda\)	\(-.234839336\)
\(-\frac{ 5}{ 3}\)	\(\frac{ 1}{ 6}\)	\(-\frac{ 25}{ 36}\)	\(-\frac{ 35}{ 72}+\frac{ 250}{ 3}\lambda\)
\(-2\)	\(0\)	\(\frac{ 1}{ 6}\)	\(100\lambda\)
\(-4\)	\(-2\)	\(-\frac{ 5}{ 3}\)	\(-.16666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666667+.969204490I\)

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

Explicit solution

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

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

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

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

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

\(\frac{ 7}{ 6}+\frac{ 96920449}{ 100000000}I\)	\(-\frac{ 4}{ 3}\)	\(\frac{ 1}{ 2}\)	\(1\)
\(\frac{ 5}{ 3}\)	\(0\)	\(-1\)	\(0\)
\(2\)	\(-4\)	\(0\)	\(0\)
\(4\)	\(0\)	\(0\)	\(0\)

\(-\frac{ 17}{ 12}-58\lambda\)	\(\frac{ 17}{ 6}\)	\(\frac{ 1}{ 2}\)	\(1\)
\(-\frac{ 25}{ 6}\)	\(0\)	\(-1\)	\(0\)
\(-8\)	\(16\)	\(0\)	\(0\)
\(-16\)	\(0\)	\(0\)	\(0\)