Calabi-Yau differential operator database v.3.0

1

New Number: 2.31 | AESZ: 7** | Superseeker: 96 -12064 | Hash: 2c494088fc85599f73fb344776dbbb28

Degree: 2

\(\theta^4-2^{4} x(2\theta+1)^2(32\theta^2+32\theta+13)+2^{16} x^{2}(2\theta+1)^2(2\theta+3)^2\)

Coefficients of the holomorphic solution: 1, 208, 107280, 70739200, 52362595600, ...
--> OEIS
Normalized instanton numbers (n₀=1): 96, -3560, -12064, 1941800, -489007584, ... ; Common denominator:...

Discriminant

\((1024z-1)^2\)

Local exponents

\(0\) \(\frac{ 1}{ 1024}\) \(\infty\)
\(0\) \(0\) \(\frac{ 1}{ 2}\)
\(0\) \(\frac{ 1}{ 4}\) \(\frac{ 1}{ 2}\)
\(0\) \(\frac{ 3}{ 4}\) \(\frac{ 3}{ 2}\)
\(0\) \(1\) \(\frac{ 3}{ 2}\)

\(0\)	\(\frac{ 1}{ 1024}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(\frac{ 1}{ 4}\)	\(\frac{ 1}{ 2}\)
\(0\)	\(\frac{ 3}{ 4}\)	\(\frac{ 3}{ 2}\)
\(0\)	\(1\)	\(\frac{ 3}{ 2}\)

Note:

This operator replaces AESZ 43, to which it is equivalent.

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

2

New Number: 2.32 | AESZ: | Superseeker: 60 307860 | Hash: c2f30268af49d0bdc6a36f8b0fce3367

Degree: 2

\(\theta^4-2^{2} 3 x(3\theta+1)(3\theta+2)(32\theta^2+32\theta+13)+2^{12} 3^{2} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 312, 268200, 297104640, 370278354600, ...
--> OEIS
Normalized instanton numbers (n₀=1): 60, -7635, 307860, -44194980, 3687387360, ... ; Common denominator:...

Discriminant

\((1728z-1)^2\)

Local exponents

\(0\) \(\frac{ 1}{ 1728}\) \(\infty\)
\(0\) \(0\) \(\frac{ 1}{ 3}\)
\(0\) \(\frac{ 1}{ 4}\) \(\frac{ 2}{ 3}\)
\(0\) \(\frac{ 3}{ 4}\) \(\frac{ 4}{ 3}\)
\(0\) \(1\) \(\frac{ 5}{ 3}\)

\(0\)	\(\frac{ 1}{ 1728}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(0\)	\(\frac{ 1}{ 4}\)	\(\frac{ 2}{ 3}\)
\(0\)	\(\frac{ 3}{ 4}\)	\(\frac{ 4}{ 3}\)
\(0\)	\(1\)	\(\frac{ 5}{ 3}\)

Note:

This is operator "2.32" from ...

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

3

New Number: 2.33 | AESZ: | Superseeker: -160 -539680 | Hash: 83a66e92381baa083f87a13e02375bc9

Degree: 2

\(\theta^4-2^{4} x(4\theta+1)(4\theta+3)(32\theta^2+32\theta+13)+2^{16} x^{2}(4\theta+1)(4\theta+3)(4\theta+5)(4\theta+7)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 624, 1251600, 3268151040, 9627237219600, ...
--> OEIS
Normalized instanton numbers (n₀=1): -160, -6920, -539680, -54568560, -6402958560, ... ; Common denominator:...

Discriminant

\((4096z-1)^2\)

Local exponents

\(0\) \(\frac{ 1}{ 4096}\) \(\infty\)
\(0\) \(0\) \(\frac{ 1}{ 4}\)
\(0\) \(\frac{ 1}{ 4}\) \(\frac{ 3}{ 4}\)
\(0\) \(\frac{ 3}{ 4}\) \(\frac{ 5}{ 4}\)
\(0\) \(1\) \(\frac{ 7}{ 4}\)

\(0\)	\(\frac{ 1}{ 4096}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(0\)	\(\frac{ 1}{ 4}\)	\(\frac{ 3}{ 4}\)
\(0\)	\(\frac{ 3}{ 4}\)	\(\frac{ 5}{ 4}\)
\(0\)	\(1\)	\(\frac{ 7}{ 4}\)

Note:

This is operator "2.33" from ...

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

4

New Number: 2.34 | AESZ: | Superseeker: -3936 -10892932064 | Hash: 3047f1b969ee26b92708d8eab03c0aed

Degree: 2

\(\theta^4-2^{4} 3 x(6\theta+1)(6\theta+5)(32\theta^2+32\theta+13)+2^{16} 3^{2} x^{2}(6\theta+1)(6\theta+5)(6\theta+7)(6\theta+11)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 3120, 41302800, 722261379840, 14304149060881680, ...
--> OEIS
Normalized instanton numbers (n₀=1): -3936, 3550992, -10892932064, 48014348072136, -264897451764337440, ... ; Common denominator:...

Discriminant

\((27648z-1)^2\)

Local exponents

\(0\) \(\frac{ 1}{ 27648}\) \(\infty\)
\(0\) \(0\) \(\frac{ 1}{ 6}\)
\(0\) \(\frac{ 1}{ 4}\) \(\frac{ 5}{ 6}\)
\(0\) \(\frac{ 3}{ 4}\) \(\frac{ 7}{ 6}\)
\(0\) \(1\) \(\frac{ 11}{ 6}\)

\(0\)	\(\frac{ 1}{ 27648}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 6}\)
\(0\)	\(\frac{ 1}{ 4}\)	\(\frac{ 5}{ 6}\)
\(0\)	\(\frac{ 3}{ 4}\)	\(\frac{ 7}{ 6}\)
\(0\)	\(1\)	\(\frac{ 11}{ 6}\)

Note:

This is operator "2.34" from ...

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

5

New Number: 3.5 | AESZ: | Superseeker: 26 103520/9 | Hash: 4ed9bc316d49a71649da0a1148f7ea9d

Degree: 3

\(\theta^4-2 x\left(102\theta^4+204\theta^3+155\theta^2+53\theta+7\right)+2^{2} x^{2}(\theta+1)^2(396\theta^2+792\theta+311)-2^{4} 7^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 14, 834, 78260, 8970850, ...
--> OEIS
Normalized instanton numbers (n₀=1): 26, 348, 103520/9, 539764, 31290280, ... ; Common denominator:...

Discriminant

\(-(196z-1)(4z-1)^2\)

Local exponents

\(0\) \(\frac{ 1}{ 196}\) \(\frac{ 1}{ 4}\) \(\infty\)
\(0\) \(0\) \(0\) \(\frac{ 1}{ 2}\)
\(0\) \(1\) \(\frac{ 1}{ 4}\) \(1\)
\(0\) \(1\) \(\frac{ 3}{ 4}\) \(2\)
\(0\) \(2\) \(1\) \(\frac{ 5}{ 2}\)

\(0\)	\(\frac{ 1}{ 196}\)	\(\frac{ 1}{ 4}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(\frac{ 1}{ 4}\)	\(1\)
\(0\)	\(1\)	\(\frac{ 3}{ 4}\)	\(2\)
\(0\)	\(2\)	\(1\)	\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to AESZ 214

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

6

New Number: 5.17 | AESZ: 119 | Superseeker: 28/3 3892/3 | Hash: dc8ec37012f2c92c83e6519935956eeb

Degree: 5

\(3^{2} \theta^4-2^{2} 3 x\left(256\theta^4+320\theta^3+271\theta^2+111\theta+18\right)+2^{7} x^{2}\left(3104\theta^4+7040\theta^3+8012\theta^2+4452\theta+927\right)-2^{15} x^{3}\left(752\theta^4+2304\theta^3+3042\theta^2+1854\theta+405\right)+2^{21} x^{4}(2\theta+1)(176\theta^3+552\theta^2+622\theta+231)-2^{31} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 24, 1128, 67200, 4634280, ...
--> OEIS
Normalized instanton numbers (n₀=1): 28/3, 394/3, 3892/3, 108262/3, 1044128, ... ; Common denominator:...

Discriminant

\(-(-1+128z)(64z-1)^2(128z-3)^2\)

Local exponents

\(0\) \(\frac{ 1}{ 128}\) \(\frac{ 1}{ 64}\) \(\frac{ 3}{ 128}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(\frac{ 1}{ 2}\)
\(0\) \(1\) \(\frac{ 1}{ 4}\) \(1\) \(1\)
\(0\) \(1\) \(\frac{ 3}{ 4}\) \(3\) \(1\)
\(0\) \(2\) \(1\) \(4\) \(\frac{ 3}{ 2}\)

\(0\)	\(\frac{ 1}{ 128}\)	\(\frac{ 1}{ 64}\)	\(\frac{ 3}{ 128}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(\frac{ 1}{ 4}\)	\(1\)	\(1\)
\(0\)	\(1\)	\(\frac{ 3}{ 4}\)	\(3\)	\(1\)
\(0\)	\(2\)	\(1\)	\(4\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.17" from ...

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

7

New Number: 5.37 | AESZ: 221 | Superseeker: 492/5 872164/5 | Hash: b7ce7a734c057660ce3d6341a7572078

Degree: 5

\(5^{2} \theta^4-2^{2} 5 x\left(404\theta^4+1096\theta^3+773\theta^2+225\theta+25\right)-2^{4} x^{2}\left(66896\theta^4+137408\theta^3+101096\theta^2+52800\theta+11625\right)-2^{8} 3 5 x^{3}(2\theta+1)(5672\theta^3+9500\theta^2+8422\theta+2689)-2^{15} 3^{2} x^{4}(2\theta+1)(1208\theta^3+2892\theta^2+2842\theta+969)-2^{20} 3^{3} x^{5}(2\theta+1)(6\theta+5)(6\theta+7)(2\theta+3)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 20, 2988, 618320, 156299500, ...
--> OEIS
Normalized instanton numbers (n₀=1): 492/5, 10376/5, 872164/5, 91316176/5, 12181916784/5, ... ; Common denominator:...

Discriminant

\(-(-1+432z)(16z+1)^2(192z+5)^2\)

Local exponents

\(-\frac{ 1}{ 16}\) \(-\frac{ 5}{ 192}\) \(0\) \(\frac{ 1}{ 432}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 4}\) \(1\) \(0\) \(1\) \(\frac{ 5}{ 6}\)
\(\frac{ 3}{ 4}\) \(3\) \(0\) \(1\) \(\frac{ 7}{ 6}\)
\(1\) \(4\) \(0\) \(2\) \(\frac{ 3}{ 2}\)

\(-\frac{ 1}{ 16}\)	\(-\frac{ 5}{ 192}\)	\(0\)	\(\frac{ 1}{ 432}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 4}\)	\(1\)	\(0\)	\(1\)	\(\frac{ 5}{ 6}\)
\(\frac{ 3}{ 4}\)	\(3\)	\(0\)	\(1\)	\(\frac{ 7}{ 6}\)
\(1\)	\(4\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.37" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-32\)	\(40\)	\(10\)	\(5\)

8

New Number: 2.71 | AESZ: | Superseeker: 0 0 | Hash: 757b011780c5986bd45a5bf434c76c28

Degree: 2

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 32, 2160, 181760, 17021200, ...
--> OEIS
Normalized instanton numbers (n₀=1): 0, -20, 0, -865, 0, ... ; Common denominator:...

Discriminant

\((-1+128z)^2\)

Local exponents

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

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

Note:

This is operator is equivalent to [2.33]. Transformation:.....

\(-\frac{ 1}{ 3}+Iarctan(\frac{ 244}{ 63})\)	\(\frac{ 2}{ 3}+136\lambda\)	\(\frac{ 1}{ 9}-\frac{ 68}{ 3}\lambda\)	\(.5042804e-2\)
\(\frac{ 2}{ 3}\)	\(\frac{ 4}{ 3}\)	\(-\frac{ 1}{ 18}\)	\(\frac{ 1}{ 9}+\frac{ 68}{ 3}\lambda\)
\(-4\)	\(2\)	\(\frac{ 4}{ 3}\)	\(\frac{ 2}{ 3}-136\lambda\)
\(-8\)	\(-4\)	\(\frac{ 2}{ 3}\)	\(-\frac{ 1}{ 3}-Iarctan(\frac{ 244}{ 63})\)

\(-.12500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000+.872284041I\)	\(\frac{ 9}{ 16}+90\lambda\)	\(\frac{ 3}{ 64}-\frac{ 15}{ 2}\lambda\)	\(.84124259e-1\)
\(\frac{ 1}{ 4}\)	\(\frac{ 9}{ 8}\)	\(-\frac{ 1}{ 96}\)	\(\frac{ 3}{ 64}+\frac{ 15}{ 2}\lambda\)
\(-3\)	\(\frac{ 3}{ 2}\)	\(\frac{ 9}{ 8}\)	\(\frac{ 9}{ 16}-90\lambda\)
\(-6\)	\(-3\)	\(\frac{ 1}{ 4}\)	\(-.12500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000-.872284041I\)

\(\frac{ 1}{ 12}+80\lambda\)	\(\frac{ 11}{ 24}+40\lambda\)	\(-\frac{ 11}{ 288}+\frac{ 10}{ 3}\lambda\)	\(.172495151\)
\(-\frac{ 1}{ 6}\)	\(\frac{ 11}{ 12}\)	\(-\frac{ 1}{ 144}\)	\(-\frac{ 11}{ 288}-\frac{ 10}{ 3}\lambda\)
\(-2\)	\(1\)	\(\frac{ 11}{ 12}\)	\(\frac{ 11}{ 24}-40\lambda\)
\(-4\)	\(-2\)	\(-\frac{ 1}{ 6}\)	\(\frac{ 1}{ 12}-80\lambda\)

\(\frac{ 7}{ 24}-44\lambda\)	\(\frac{ 17}{ 48}-22\lambda\)	\(-\frac{ 119}{ 576}-\frac{ 77}{ 6}\lambda\)	\(.228135608\)
\(-\frac{ 7}{ 12}\)	\(\frac{ 17}{ 24}\)	\(-\frac{ 49}{ 288}\)	\(-\frac{ 119}{ 576}+\frac{ 77}{ 6}\lambda\)
\(-1\)	\(\frac{ 1}{ 2}\)	\(\frac{ 17}{ 24}\)	\(\frac{ 17}{ 48}+22\lambda\)
\(-2\)	\(-1\)	\(-\frac{ 7}{ 12}\)	\(\frac{ 7}{ 24}+44\lambda\)

\(\frac{ 4}{ 3}-\frac{ 1708656527755053726527184603581775358292629291158824225586117747415508815310392346494291960809911989}{ 1296284848710416017198988117968146871699058960076991381465223285947803281706700313960088400294610326}I\)	\(\frac{ 4}{ 3}\)	\(\frac{ 1}{ 2}\)	\(1\)
\(-\frac{ 2}{ 3}\)	\(0\)	\(-1\)	\(0\)
\(4\)	\(-8\)	\(0\)	\(0\)
\(8\)	\(0\)	\(0\)	\(0\)

\(\frac{ 9}{ 8}-\frac{ 872284041}{ 1000000000}I\)	\(\frac{ 3}{ 4}\)	\(\frac{ 1}{ 2}\)	\(1\)
\(-\frac{ 1}{ 4}\)	\(0\)	\(-1\)	\(0\)
\(3\)	\(-6\)	\(0\)	\(0\)
\(6\)	\(0\)	\(0\)	\(0\)

\(\frac{ 11}{ 12}-80\lambda\)	\(\frac{ 1}{ 6}\)	\(\frac{ 1}{ 2}\)	\(1\)
\(\frac{ 1}{ 6}\)	\(0\)	\(-1\)	\(0\)
\(2\)	\(-4\)	\(0\)	\(0\)
\(4\)	\(0\)	\(0\)	\(0\)

\(\frac{ 17}{ 24}+44\lambda\)	\(-\frac{ 5}{ 12}\)	\(\frac{ 1}{ 2}\)	\(1\)
\(\frac{ 7}{ 12}\)	\(0\)	\(-1\)	\(0\)
\(1\)	\(-2\)	\(0\)	\(0\)
\(2\)	\(0\)	\(0\)	\(0\)

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Note:

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:

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:

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:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice